Title:	Probability Distributions as S3 Objects
Version:	0.2.2
Description:	Tools to create and manipulate probability distributions using S3. Generics pdf(), cdf(), quantile(), and random() provide replacements for base R's d/p/q/r style functions. Functions and arguments have been named carefully to minimize confusion for students in intro stats courses. The documentation for each distribution contains detailed mathematical notes.
License:	MIT + file LICENSE
URL:	https://github.com/alexpghayes/distributions3, https://alexpghayes.github.io/distributions3/
BugReports:	https://github.com/alexpghayes/distributions3/issues
Imports:	ggplot2, glue, rlang
Suggests:	cowplot, knitr, PoissonBinomial, revdbayes (≥ 1.3.5), rmarkdown, testthat (≥ 3.0.0), tibble, vctrs
VignetteBuilder:	knitr
Encoding:	UTF-8
RoxygenNote:	7.3.2
Config/testthat/edition:	3
Config/testthat/parallel:	true
NeedsCompilation:	no
Packaged:	2024-09-16 15:51:35 UTC; alex
Author:	Alex Hayes [aut, cre], Ralph Moller-Trane [aut], Emil Hvitfeldt [ctb], Daniel Jordan [aut], Paul Northrop [aut], Moritz N. Lang [aut], Achim Zeileis [aut], Bruna Wundervald [ctb], Alessandro Gasparini [ctb]
Maintainer:	Alex Hayes <alexpghayes@gmail.com>
Repository:	CRAN
Date/Publication:	2024-09-16 16:20:02 UTC

distributions3: Probability Distributions as S3 Objects

Description

Tools to create and manipulate probability distributions using S3. Generics pdf(), cdf(), quantile(), and random() provide replacements for base R's d/p/q/r style functions. Functions and arguments have been named carefully to minimize confusion for students in intro stats courses. The documentation for each distribution contains detailed mathematical notes.

Author(s)

Maintainer: Alex Hayes alexpghayes@gmail.com (ORCID)

Authors:

• Ralph Moller-Trane

• Daniel Jordan dandermotj@gmail.com

• Paul Northrop p.northrop@ucl.ac.uk

• Moritz N. Lang moritz.n.lang@gmail.com (ORCID)

• Achim Zeileis Achim.Zeileis@R-project.org (ORCID)

Other contributors:

• Emil Hvitfeldt emilhhvitfeldt@gmail.com (ORCID) [contributor]

• Bruna Wundervald brunadaviesw@gmail.com [contributor]

• Alessandro Gasparini alessandro.gasparini@ki.se [contributor]

See Also

Useful links:

• https://github.com/alexpghayes/distributions3

• https://alexpghayes.github.io/distributions3/

• Report bugs at https://github.com/alexpghayes/distributions3/issues

Utilities for distributions3 objects

Description

Various utility functions to implement methods for distributions with a unified workflow, in particular to facilitate working with vectorized distributions3 objects. These are particularly useful in the computation of densities, probabilities, quantiles, and random samples when classical d/p/q/r functions are readily available for the distribution of interest.

Usage

apply_dpqr(d, FUN, at, elementwise = NULL, drop = TRUE, type = NULL, ...)

make_support(min, max, d, drop = TRUE)

make_positive_integer(n)

Arguments

Examples

## Implementing a new distribution based on the provided utility functions
## Illustration: Gaussian distribution
## Note: Gaussian() is really just a copy of Normal() with a different class/distribution name


## Generator function for the distribution object.
Gaussian <- function(mu = 0, sigma = 1) {
  stopifnot(
    "parameter lengths do not match (only scalars are allowed to be recycled)" =
      length(mu) == length(sigma) | length(mu) == 1 | length(sigma) == 1
  )
  d <- data.frame(mu = mu, sigma = sigma)
  class(d) <- c("Gaussian", "distribution")
  d
}

## Set up a vector Y containing four Gaussian distributions:
Y <- Gaussian(mu = 1:4, sigma = c(1, 1, 2, 2))
Y

## Extract the underlying parameters:
as.matrix(Y)


## Extractor functions for moments of the distribution include
## mean(), variance(), skewness(), kurtosis().
## These can be typically be defined as functions of the list of parameters.
mean.Gaussian <- function(x, ...) {
  rlang::check_dots_used()
  setNames(x$mu, names(x))
}
## Analogously for other moments, see distributions3:::variance.Normal etc.

mean(Y)


## The support() method should return a matrix of "min" and "max" for the
## distribution. The make_support() function helps to set the right names and
## dimension.
support.Gaussian <- function(d, drop = TRUE, ...) {
  min <- rep(-Inf, length(d))
  max <- rep(Inf, length(d))
  make_support(min, max, d, drop = drop)
}

support(Y)


## Evaluating certain functions associated with the distribution, e.g.,
## pdf(), log_pdf(), cdf() quantile(), random(), etc. The apply_dpqr()
## function helps to call the typical d/p/q/r functions (like dnorm,
## pnorm, etc.) and set suitable names and dimension.
pdf.Gaussian <- function(d, x, elementwise = NULL, drop = TRUE, ...) {
  FUN <- function(at, d) dnorm(x = at, mean = d$mu, sd = d$sigma, ...)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "density", elementwise = elementwise, drop = drop)
}

## Evaluate all densities at the same argument (returns vector):
pdf(Y, 0)

## Evaluate all densities at several arguments (returns matrix):
pdf(Y, c(0, 5))

## Evaluate each density at a different argument (returns vector):
pdf(Y, 4:1)

## Force evaluation of each density at a different argument (returns vector)
## or at all arguments (returns matrix):
pdf(Y, 4:1, elementwise = TRUE)
pdf(Y, 4:1, elementwise = FALSE)

## Drawing random() samples also uses apply_dpqr() with the argument
## n assured to be a positive integer.
random.Gaussian <- function(x, n = 1L, drop = TRUE, ...) {
  n <- make_positive_integer(n)
  if (n == 0L) {
    return(numeric(0L))
  }
  FUN <- function(at, d) rnorm(n = at, mean = d$mu, sd = d$sigma)
  apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}

## One random sample for each distribution (returns vector):
random(Y, 1)

## Several random samples for each distribution (returns matrix):
random(Y, 3)


## For further analogous methods see the "Normal" distribution provided
## in distributions3.
methods(class = "Normal")

Create a Bernoulli distribution

Description

Bernoulli distributions are used to represent events like coin flips when there is single trial that is either successful or unsuccessful. The Bernoulli distribution is a special case of the Binomial() distribution with n = 1.

Usage

Bernoulli(p = 0.5)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Bernoulli random variable with parameter p = p. Some textbooks also define q = 1 - p, or use \pi instead of p.

The Bernoulli probability distribution is widely used to model binary variables, such as 'failure' and 'success'. The most typical example is the flip of a coin, when p is thought as the probability of flipping a head, and q = 1 - p is the probability of flipping a tail.

Support: \{0, 1\}

Mean: p

Variance: p \cdot (1 - p) = p \cdot q

Probability mass function (p.m.f):

P(X = x) = p^x (1 - p)^{1-x} = p^x q^{1-x}

Cumulative distribution function (c.d.f):

P(X \le x) = \left \{ \begin{array}{ll} 0 & x < 0 \\ 1 - p & 0 \leq x < 1 \\ 1 & x \geq 1 \end{array} \right.

Moment generating function (m.g.f):

E(e^{tX}) = (1 - p) + p e^t

Value

A Bernoulli object.

See Also

Other discrete distributions: Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Beta distribution

Description

Create a Beta distribution

Usage

Beta(alpha = 1, beta = 1)

Arguments

Value

A beta object.

See Also

Other continuous distributions: Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Binomial distribution

Description

Binomial distributions are used to represent situations can that can be thought as the result of n Bernoulli experiments (here the n is defined as the size of the experiment). The classical example is n independent coin flips, where each coin flip has probability p of success. In this case, the individual probability of flipping heads or tails is given by the Bernoulli(p) distribution, and the probability of having x equal results (x heads, for example), in n trials is given by the Binomial(n, p) distribution. The equation of the Binomial distribution is directly derived from the equation of the Bernoulli distribution.

Usage

Binomial(size, p = 0.5)

Arguments

Details

The Binomial distribution comes up when you are interested in the portion of people who do a thing. The Binomial distribution also comes up in the sign test, sometimes called the Binomial test (see stats::binom.test()), where you may need the Binomial C.D.F. to compute p-values.

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Binomial random variable with parameter size = n and p = p. Some textbooks define q = 1 - p, or called \pi instead of p.

Support: \{0, 1, 2, ..., n\}

Mean: np

Variance: np \cdot (1 - p) = np \cdot q

Probability mass function (p.m.f):

P(X = k) = {n \choose k} p^k (1 - p)^{n-k}

Cumulative distribution function (c.d.f):

P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n \choose i} p^i (1 - p)^{n-i}

Moment generating function (m.g.f):

E(e^{tX}) = (1 - p + p e^t)^n

Value

A Binomial object.

See Also

Other discrete distributions: Bernoulli(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create a Categorical distribution

Description

Create a Categorical distribution

Usage

Categorical(outcomes, p = NULL)

Arguments

Value

A Categorical object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Create a Cauchy distribution

Description

Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.

Usage

Cauchy(location = 0, scale = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Cauchy variable with mean ⁠location =⁠ x_0 and scale = \gamma.

Support: R, the set of all real numbers

Mean: Undefined.

Variance: Undefined.

Probability density function (p.d.f):

f(x) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma} \right)^2 \right]}

Cumulative distribution function (c.d.f):

F(t) = \frac{1}{\pi} \arctan \left( \frac{t - x_0}{\gamma} \right) + \frac{1}{2}

Moment generating function (m.g.f):

Does not exist.

Value

A Cauchy object.

See Also

Other continuous distributions: Beta(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a probability distribution

Description

Generic function for computing probabilities from distribution objects based on the cumulative distribution function (CDF).

Usage

cdf(d, x, drop = TRUE, ...)

Arguments

Value

Probabilities corresponding to the vector x.

Examples

## distribution object
X <- Normal()
## probabilities from CDF
cdf(X, c(1, 2, 3, 4, 5))

Evaluate the cumulative distribution function of a Bernoulli distribution

Description

Evaluate the cumulative distribution function of a Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Beta distribution

Description

Evaluate the cumulative distribution function of a Beta distribution

Usage

## S3 method for class 'Beta'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Binomial distribution

Description

Evaluate the cumulative distribution function of a Binomial distribution

Usage

## S3 method for class 'Binomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Categorical distribution

Description

Evaluate the cumulative distribution function of a Categorical distribution

Usage

## S3 method for class 'Categorical'
cdf(d, x, ...)

Arguments

Value

A vector of probabilities, one for each element of x.

Examples

set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Evaluate the cumulative distribution function of a Cauchy distribution

Description

Evaluate the cumulative distribution function of a Cauchy distribution

Usage

## S3 method for class 'Cauchy'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a chi square distribution

Description

Evaluate the cumulative distribution function of a chi square distribution

Usage

## S3 method for class 'ChiSquare'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an Erlang distribution

Description

Evaluate the cumulative distribution function of an Erlang distribution

Usage

## S3 method for class 'Erlang'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an Exponential distribution

Description

Evaluate the cumulative distribution function of an Exponential distribution

Usage

## S3 method for class 'Exponential'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an F distribution

Description

Evaluate the cumulative distribution function of an F distribution

Usage

## S3 method for class 'FisherF'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Frechet distribution

Description

Evaluate the cumulative distribution function of a Frechet distribution

Usage

## S3 method for class 'Frechet'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Gamma distribution

Description

Evaluate the cumulative distribution function of a Gamma distribution

Usage

## S3 method for class 'Gamma'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Geometric distribution

Description

Evaluate the cumulative distribution function of a Geometric distribution

Usage

## S3 method for class 'Geometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Geometric distribution: pdf.Geometric(), quantile.Geometric(), random.Geometric()

Examples

set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a GEV distribution

Description

Evaluate the cumulative distribution function of a GEV distribution

Usage

## S3 method for class 'GEV'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a GP distribution

Description

Evaluate the cumulative distribution function of a GP distribution

Usage

## S3 method for class 'GP'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Gumbel distribution

Description

Evaluate the cumulative distribution function of a Gumbel distribution

Usage

## S3 method for class 'Gumbel'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a hurdle negative binomial distribution

Description

Evaluate the cumulative distribution function of a hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a hurdle Poisson distribution

Description

Evaluate the cumulative distribution function of a hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a HyperGeometric distribution

Description

Evaluate the cumulative distribution function of a HyperGeometric distribution

Usage

## S3 method for class 'HyperGeometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other HyperGeometric distribution: pdf.HyperGeometric(), quantile.HyperGeometric(), random.HyperGeometric()

Examples

set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a Logistic distribution

Description

Evaluate the cumulative distribution function of a Logistic distribution

Usage

## S3 method for class 'Logistic'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Logistic distribution: pdf.Logistic(), quantile.Logistic(), random.Logistic()

Examples

set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a LogNormal distribution

Description

Evaluate the cumulative distribution function of a LogNormal distribution

Usage

## S3 method for class 'LogNormal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other LogNormal distribution: fit_mle.LogNormal(), pdf.LogNormal(), quantile.LogNormal(), random.LogNormal()

Examples

set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a negative binomial distribution

Description

Evaluate the cumulative distribution function of a negative binomial distribution

Usage

## S3 method for class 'NegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other NegativeBinomial distribution: pdf.NegativeBinomial(), quantile.NegativeBinomial(), random.NegativeBinomial()

Examples

set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Evaluate the cumulative distribution function of a Normal distribution

Description

Evaluate the cumulative distribution function of a Normal distribution

Usage

## S3 method for class 'Normal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Normal distribution: fit_mle.Normal(), pdf.Normal(), quantile.Normal()

Examples

set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Poisson distribution

Description

Evaluate the cumulative distribution function of a Poisson distribution

Usage

## S3 method for class 'Poisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a PoissonBinomial distribution

Description

Evaluate the cumulative distribution function of a PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
cdf(
  d,
  x,
  drop = TRUE,
  elementwise = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  verbose = TRUE,
  ...
)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Evaluate the cumulative distribution function of an RevWeibull distribution

Description

Evaluate the cumulative distribution function of an RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a StudentsT distribution

Description

Evaluate the cumulative distribution function of a StudentsT distribution

Usage

## S3 method for class 'StudentsT'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other StudentsT distribution: pdf.StudentsT(), quantile.StudentsT(), random.StudentsT()

Examples

set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Evaluate the cumulative distribution function of a Tukey distribution

Description

Evaluate the cumulative distribution function of a Tukey distribution

Usage

## S3 method for class 'Tukey'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Tukey distribution: quantile.Tukey()

Examples

set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a continuous Uniform distribution

Description

Evaluate the cumulative distribution function of a continuous Uniform distribution

Usage

## S3 method for class 'Uniform'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Weibull distribution

Description

Evaluate the cumulative distribution function of a Weibull distribution

Usage

## S3 method for class 'Weibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Weibull distribution: pdf.Weibull(), quantile.Weibull(), random.Weibull()

Examples

set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a zero-inflated negative binomial distribution

Description

Evaluate the cumulative distribution function of a zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-inflated Poisson distribution

Description

Evaluate the cumulative distribution function of a zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-truncated negative binomial distribution

Description

Evaluate the cumulative distribution function of a zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-truncated Poisson distribution

Description

Evaluate the cumulative distribution function of a zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a Chi-Square distribution

Description

Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.

Usage

ChiSquare(df)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a \chi^2 random variable with df = k.

Support: R^+, the set of positive real numbers

Mean: k

Variance: 2k

Probability density function (p.d.f):

f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard normal is sometimes called the "error function". The notation \Phi(t) also stands for the c.d.f. of a standard normal evaluated at t. Z-tables list the value of \Phi(t) for various t.

Moment generating function (m.g.f):

E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}

Value

A ChiSquare object.

Transformations

A squared standard Normal() distribution is equivalent to a \chi^2_1 distribution with one degree of freedom. The \chi^2 distribution is a special case of the Gamma() distribution with shape (TODO: check this) parameter equal to a half. Sums of \chi^2 distributions are also distributed as \chi^2 distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two \chi^2 distributions is a FisherF() distribution. The ratio of a Normal() and the square root of a scaled ChiSquare() is a StudentsT() distribution.

See Also

Other continuous distributions: Beta(), Cauchy(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

The hurdle negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-hurdle negative binomial distribution with parameters mu, theta (or size), and pi.

Usage

dhnbinom(x, mu, theta, size, pi, log = FALSE)

phnbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qhnbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rhnbinom(n, mu, theta, size, pi)

Arguments

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four hnbinom functions for the hurdle negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

Note, however, that the precision of qhnbinom for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

See Also

HurdleNegativeBinomial, dnbinom

Examples

## theoretical probabilities for a hurdle negative binomial distribution
x <- 0:8
p <- dhnbinom(x, mu = 2.5, theta = 1, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhnbinom(500, mu = 2.5, theta = 1, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

The hurdle Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-hurdle Poisson distribution with parameters lambda and pi.

Usage

dhpois(x, lambda, pi, log = FALSE)

phpois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qhpois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rhpois(n, lambda, pi)

Arguments

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four hpois functions for the hurdle Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

Note, however, that the precision of qhpois for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

See Also

HurdlePoisson, dpois

Examples

## theoretical probabilities for a hurdle Poisson distribution
x <- 0:8
p <- dhpois(x, lambda = 2.5, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhpois(500, lambda = 2.5, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

The zero-inflated negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-inflated negative binomial distribution with parameters mu, theta (or size), and pi.

Usage

dzinbinom(x, mu, theta, size, pi, log = FALSE)

pzinbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qzinbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rzinbinom(n, mu, theta, size, pi)

Arguments

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four zinbinom functions for the zero-inflated negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

Note, however, that the precision of qzinbinom for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

See Also

ZINegativeBinomial, dnbinom

Examples

## theoretical probabilities for a zero-inflated negative binomial distribution
x <- 0:8
p <- dzinbinom(x, mu = 2.5, theta = 1, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzinbinom(500, mu = 2.5, theta = 1, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

The zero-inflated Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-inflated Poisson distribution with parameters lambda and pi.

Usage

dzipois(x, lambda, pi, log = FALSE)

pzipois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qzipois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rzipois(n, lambda, pi)

Arguments

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four zipois functions for the zero-inflated Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

Note, however, that the precision of qzipois for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

See Also

ZIPoisson, dpois

Examples

## theoretical probabilities for a zero-inflated Poisson distribution
x <- 0:8
p <- dzipois(x, lambda = 2.5, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzipois(500, lambda = 2.5, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

The zero-truncated negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-truncated negative binomial distribution with parameters mu and theta (or size).

Usage

dztnbinom(x, mu, theta, size, log = FALSE)

pztnbinom(q, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

qztnbinom(p, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

rztnbinom(n, mu, theta, size)

Arguments

Details

The negative binomial distribution left-truncated at zero (or zero-truncated negative binomial for short) is the distribution obtained, when considering a negative binomial variable Y conditional on Y being greater than zero.

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four ztnbinom functions for the zero-truncated negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

See Also

ZTNegativeBinomial, dnbinom

Examples

## theoretical probabilities for a zero-truncated negative binomial distribution
x <- 0:8
p <- dztnbinom(x, mu = 2.5, theta = 1)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztnbinom(500, mu = 2.5, theta = 1)
hist(y, breaks = -1:max(y) + 0.5)

The zero-truncated Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-truncated Poisson distribution with parameter lambda.

Usage

dztpois(x, lambda, log = FALSE)

pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)

qztpois(p, lambda, lower.tail = TRUE, log.p = FALSE)

rztpois(n, lambda)

Arguments

Details

The Poisson distribution left-truncated at zero (or zero-truncated Poisson for short) is the distribution obtained, when considering a Poisson variable Y conditional on Y being greater than zero.

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four ztpois functions for the zero-truncated Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

See Also

ZTPoisson, dpois

Examples

## theoretical probabilities for a zero-truncated Poisson distribution
x <- 0:8
p <- dztpois(x, lambda = 2.5)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztpois(500, lambda = 2.5)
hist(y, breaks = -1:max(y) + 0.5)

Create an Erlang distribution

Description

The Erlang distribution is a two-parameter family of continuous probability distributions with support x \in [0,\infty). The two parameters are a positive integer shape parameter k and a positive real rate parameter \lambda. The Erlang distribution with shape parameter k = 1 simplifies to the exponential distribution, and it is a special case of the gamma distribution. It corresponds to a sum of k independent exponential variables with mean 1 / \lambda each.

Usage

Erlang(k, lambda)

Arguments

Value

An Erlang object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create an Exponential distribution

Description

Exponential distributions are frequently used for modeling the amount of time that passes until a specific event occurs. For example, exponential distributions could be used to model the time between two earthquakes, the amount of delay between internet packets, or the amount of time a piece of machinery can run before needing repair.

Usage

Exponential(rate = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be an Exponential random variable with rate parameter rate = \lambda.

Support: x \in (0, \infty)

Mean: \frac{1}{\lambda}

Variance: \frac{1}{\lambda^2}

Probability density function (p.d.f):

f(x) = \lambda e^{-\lambda x}

Cumulative distribution function (c.d.f):

F(x) = 1 - e^{-\lambda x}

Moment generating function (m.g.f):

\frac{\lambda}{\lambda - t}, for t < \lambda

Value

An Exponential object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Goals scored in all 2018 FIFA World Cup matches

Description

Data from all 64 matches in the 2018 FIFA World Cup along with predicted ability differences based on bookmakers odds.

Usage

data("FIFA2018", package = "distributions3")

Format

A data frame with 128 rows and 7 columns.

goals

integer. Number of goals scored in normal time (90 minutes), \ i.e., excluding potential extra time or penalties in knockout matches.

team

character. 3-letter FIFA code for the team.

match

integer. Match ID ranging from 1 (opening match) to 64 (final).

type

factor. Type of match for groups A to H, round of 16 (R16), quarter final, semi-final, match for 3rd place, and final.

stage

factor. Group vs. knockout tournament stage.

logability

numeric. Estimated log-ability for each team based on bookmaker consensus model.

difference

numeric. Difference in estimated log-abilities between a team and its opponent in each match.

Details

To investigate the number of goals scored per match in the 2018 FIFA World Cup, FIFA2018 provides two rows, one for each team, for each of the matches during the tournament. In addition some basic meta-information for the matches (an ID, team name abbreviations, type of match, group vs. knockout stage), information on the estimated log-ability for each team is provided. These have been estimated by Zeileis et al. (2018) prior to the start of the tournament (2018-05-20) based on quoted odds from 26 online bookmakers using the bookmaker consensus model of Leitner et al. (2010). The difference in log-ability between a team and its opponent is a useful predictor for the number of goals scored.

To model the data a basic Poisson regression model provides a good fit. This treats the number of goals by the two teams as independent given the ability difference which is a reasonable assumption in this data set.

Source

The goals for each match have been obtained from Wikipedia (https://en.wikipedia.org/wiki/2018_FIFA_World_Cup) and the log-abilities from Zeileis et al. (2018) based on quoted odds from Oddschecker.com and Bwin.com.

References

Leitner C, Zeileis A, Hornik K (2010). Forecasting Sports Tournaments by Ratings of (Prob)abilities: A Comparison for the EURO 2008. International Journal of Forecasting, 26(3), 471-481. doi:10.1016/j.ijforecast.2009.10.001

Zeileis A, Leitner C, Hornik K (2018). Probabilistic Forecasts for the 2018 FIFA World Cup Based on the Bookmaker Consensus Model. Working Paper 2018-09, Working Papers in Economics and Statistics, Research Platform Empirical and Experimental Economics, University of Innsbruck. https://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09

Examples

## load data
data("FIFA2018", package = "distributions3")

## observed relative frequencies of goals in all matches
obsrvd <- prop.table(table(FIFA2018$goals))

## expected probabilities assuming a simple Poisson model,
## using the average number of goals across all teams/matches
## as the point estimate for the mean (lambda) of the distribution
p_const <- Poisson(lambda = mean(FIFA2018$goals))
p_const
expctd <- pdf(p_const, 0:6)

## comparison: observed vs. expected frequencies
## frequencies for 3 and 4 goals are slightly overfitted
## while 5 and 6 goals are slightly underfitted
cbind("observed" = obsrvd, "expected" = expctd)

## instead of fitting the same average Poisson model to all
## teams/matches, take ability differences into account
m <- glm(goals ~ difference, data = FIFA2018, family = poisson)
summary(m)
## when the ratio of abilities increases by 1 percent, the
## expected number of goals increases by around 0.4 percent

## this yields a different predicted Poisson distribution for
## each team/match
p_reg <- Poisson(lambda = fitted(m))
head(p_reg)

## as an illustration, the following goal distributions
## were expected for the final (that France won 4-2 against Croatia)
p_final <- tail(p_reg, 2)
p_final
pdf(p_final, 0:6)
## clearly France was expected to score more goals than Croatia
## but both teams scored more goals than expected, albeit not unlikely many

## assuming independence of the number of goals scored, obtain
## table of possible match results (after normal time), along with
## overall probabilities of win/draw/lose
res <- outer(pdf(p_final[1], 0:6), pdf(p_final[2], 0:6))
sum(res[lower.tri(res)]) ## France wins
sum(diag(res))           ## draw
sum(res[upper.tri(res)]) ## France loses

## update expected frequencies table based on regression model
expctd <- pdf(p_reg, 0:6)
head(expctd)
expctd <- colMeans(expctd)
cbind("observed" = obsrvd, "expected" = expctd)

Create an F distribution

Description

Create an F distribution

Usage

FisherF(df1, df2, lambda = 0)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

TODO

Value

A FisherF object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Fit a distribution to data

Description

Generic function for fitting maximum-likelihood estimates (MLEs) of a distribution based on empirical data.

Usage

fit_mle(d, x, ...)

Arguments

Value

A distribution (the same kind as d) where the parameters are the MLE estimates based on x.

Examples

X <- Normal()
fit_mle(X, c(-1, 0, 0, 0, 3))

Fit a Bernoulli distribution to data

Description

Fit a Bernoulli distribution to data

Usage

## S3 method for class 'Bernoulli'
fit_mle(d, x, ...)

Arguments

Value

a Bernoulli object

Fit a Binomial distribution to data

Description

The fit distribution will inherit the same size parameter as the Binomial object passed.

Usage

## S3 method for class 'Binomial'
fit_mle(d, x, ...)

Arguments

Value

a Binomial object

Fit an Exponential distribution to data

Description

Fit an Exponential distribution to data

Usage

## S3 method for class 'Exponential'
fit_mle(d, x, ...)

Arguments

Value

An Exponential object.

Fit a Gamma distribution to data

Description

Fit a Gamma distribution to data

Usage

## S3 method for class 'Gamma'
fit_mle(d, x, ...)

Arguments

Value

a Gamma object

Fit a Geometric distribution to data

Description

Fit a Geometric distribution to data

Usage

## S3 method for class 'Geometric'
fit_mle(d, x, ...)

Arguments

Value

a Geometric object

Fit a Log Normal distribution to data

Description

Fit a Log Normal distribution to data

Usage

## S3 method for class 'LogNormal'
fit_mle(d, x, ...)

Arguments

Value

A LogNormal object.

See Also

Other LogNormal distribution: cdf.LogNormal(), pdf.LogNormal(), quantile.LogNormal(), random.LogNormal()

Fit a Normal distribution to data

Description

Fit a Normal distribution to data

Usage

## S3 method for class 'Normal'
fit_mle(d, x, ...)

Arguments

Value

A Normal object.

See Also

Other Normal distribution: cdf.Normal(), pdf.Normal(), quantile.Normal()

Fit an Poisson distribution to data

Description

Fit an Poisson distribution to data

Usage

## S3 method for class 'Poisson'
fit_mle(d, x, ...)

Arguments

Value

An Poisson object.

Create a Frechet distribution

Description

The Frechet distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter \xi is positive. It may be referred to as a type II extreme value distribution.

Usage

Frechet(location = 0, scale = 1, shape = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Frechet random variable with location parameter location = m, scale parameter scale = s, and shape parameter shape = \alpha. A Frechet(m, s, \alpha) distribution is equivalent to a ⁠\link{GEV}⁠(m + s, s / \alpha, 1 / \alpha) distribution.

Support: (m, \infty).

Mean: m + s\Gamma(1 - 1/\alpha), for \alpha > 1; undefined otherwise.

Median: m + s(\ln 2)^{-1/\alpha}.

Variance: s^2 [\Gamma(1 - 2 / \alpha) - \Gamma(1 - 1 / \alpha)^2] for \alpha > 2; undefined otherwise.

Probability density function (p.d.f):

f(x) = \alpha s ^ {-1} [(x - m) / s] ^ {-(1 + \alpha)}% \exp\{-[(x - m) / s] ^ {-\alpha} \}

for x > m. The p.d.f. is 0 for x \leq m.

Cumulative distribution function (c.d.f):

F(x) = \exp\{-[(x - m) / s] ^ {-\alpha} \}

for x > m. The c.d.f. is 0 for x \leq m.

Value

A Frechet object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Gamma distribution

Description

Several important distributions are special cases of the Gamma distribution. When the shape parameter is 1, the Gamma is an exponential distribution with parameter 1/\beta. When the shape = n/2 and rate = 1/2, the Gamma is a equivalent to a chi squared distribution with n degrees of freedom. Moreover, if we have X_1 is Gamma(\alpha_1, \beta) and X_2 is Gamma(\alpha_2, \beta), a function of these two variables of the form \frac{X_1}{X_1 + X_2} Beta(\alpha_1, \alpha_2). This last property frequently appears in another distributions, and it has extensively been used in multivariate methods. More about the Gamma distribution will be added soon.

Usage

Gamma(shape, rate = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Gamma random variable with parameters shape = \alpha and rate = \beta.

Support: x \in (0, \infty)

Mean: \frac{\alpha}{\beta}

Variance: \frac{\alpha}{\beta^2}

Probability density function (p.m.f):

f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}

Cumulative distribution function (c.d.f):

f(x) = \frac{\Gamma(\alpha, \beta x)}{\Gamma{\alpha}}

Moment generating function (m.g.f):

E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta

Value

A Gamma object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create a Geometric distribution

Description

The Geometric distribution can be thought of as a generalization of the Bernoulli() distribution where we ask: "if I keep flipping a coin with probability p of heads, what is the probability I need k flips before I get my first heads?" The Geometric distribution is a special case of Negative Binomial distribution.

Usage

Geometric(p = 0.5)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Geometric random variable with success probability p = p. Note that there are multiple parameterizations of the Geometric distribution.

Support: 0 < p < 1, x = 0, 1, \dots

Mean: \frac{1-p}{p}

Variance: \frac{1-p}{p^2}

Probability mass function (p.m.f):

P(X = x) = p(1-p)^x,

Cumulative distribution function (c.d.f):

P(X \le x) = 1 - (1-p)^{x+1}

Moment generating function (m.g.f):

E(e^{tX}) = \frac{pe^t}{1 - (1-p)e^t}

Value

A Geometric object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a Generalised Extreme Value (GEV) distribution

Description

The GEV distribution arises from the Extremal Types Theorem, which is rather like the Central Limit Theorem (see ⁠\link{Normal}⁠) but it relates to the maximum of n i.i.d. random variables rather than to the sum. If, after a suitable linear rescaling, the distribution of this maximum tends to a non-degenerate limit as n tends to infinity then this limit must be a GEV distribution. The requirement that the variables are independent can be relaxed substantially. Therefore, the GEV distribution is often used to model the maximum of a large number of random variables.

Usage

GEV(mu = 0, sigma = 1, xi = 0)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a GEV random variable with location parameter mu = \mu, scale parameter sigma = \sigma and shape parameter xi = \xi.

Support: (-\infty, \mu - \sigma / \xi) for \xi < 0; (\mu - \sigma / \xi, \infty) for \xi > 0; and R, the set of all real numbers, for \xi = 0.

Mean: \mu + \sigma[\Gamma(1 - \xi) - 1]/\xi for \xi < 1, \xi \neq 0; \mu + \sigma\gamma for \xi = 0, where \gamma is Euler's constant, approximately equal to 0.57722; undefined otherwise.

Median: \mu + \sigma[(\ln 2) ^ {-\xi} - 1]/\xi for \xi \neq 0; \mu - \sigma\ln(\ln 2) for \xi = 0.

Variance: \sigma^2 [\Gamma(1 - 2 \xi) - \Gamma(1 - \xi)^2] / \xi^2 for \xi < 1 / 2, \xi \neq 0; \sigma^2 \pi^2 / 6 for \xi = 0; undefined otherwise.

Probability density function (p.d.f):

If \xi \neq 0 then

f(x) = \sigma ^ {-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}% \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}

for 1 + \xi (x - \mu) / \sigma > 0. The p.d.f. is 0 outside the support.

In the \xi = 0 (Gumbel) special case

f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}

for x in R, the set of all real numbers.

Cumulative distribution function (c.d.f):

If \xi \neq 0 then

F(x) = \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}

for 1 + \xi (x - \mu) / \sigma > 0. The c.d.f. is 0 below the support and 1 above the support.

In the \xi = 0 (Gumbel) special case

F(x) = \exp\{-\exp[-(x - \mu) / \sigma] \}

for x in R, the set of all real numbers.

Value

A GEV object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Generalised Pareto (GP) distribution

Description

The GP distribution has a link to the ⁠\link{GEV}⁠ distribution. Suppose that the maximum of n i.i.d. random variables has approximately a GEV distribution. For a sufficiently large threshold u, the conditional distribution of the amount (the threshold excess) by which a variable exceeds u given that it exceeds u has approximately a GP distribution. Therefore, the GP distribution is often used to model the threshold excesses of a high threshold u. The requirement that the variables are independent can be relaxed substantially, but then exceedances of u may cluster.

Usage

GP(mu = 0, sigma = 1, xi = 0)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a GP random variable with location parameter mu = \mu, scale parameter sigma = \sigma and shape parameter xi = \xi.

Support: [\mu, \mu - \sigma / \xi] for \xi < 0; [\mu, \infty) for \xi \geq 0.

Mean: \mu + \sigma/(1 - \xi) for \xi < 1; undefined otherwise.

Median: \mu + \sigma[2 ^ \xi - 1]/\xi for \xi \neq 0; \mu + \sigma\ln 2 for \xi = 0.

Variance: \sigma^2 / (1 - \xi)^2 (1 - 2\xi) for \xi < 1 / 2; undefined otherwise.

Probability density function (p.d.f):

If \xi \neq 0 then

f(x) = \sigma^{-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}

for 1 + \xi (x - \mu) / \sigma > 0. The p.d.f. is 0 outside the support.

In the \xi = 0 special case

f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]

for x in [\mu, \infty). The p.d.f. is 0 outside the support.

Cumulative distribution function (c.d.f):

If \xi \neq 0 then

F(x) = 1 - \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}

for 1 + \xi (x - \mu) / \sigma > 0. The c.d.f. is 0 below the support and 1 above the support.

In the \xi = 0 special case

F(x) = 1 - \exp[-(x - \mu) / \sigma] \}

for x in R, the set of all real numbers.

Value

A GP object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Gumbel distribution

Description

The Gumbel distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter \xi is equal to 0. It may be referred to as a type I extreme value distribution.

Usage

Gumbel(mu = 0, sigma = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Gumbel random variable with location parameter mu = \mu, scale parameter sigma = \sigma.

Support: R, the set of all real numbers.

Mean: \mu + \sigma\gamma, where \gamma is Euler's constant, approximately equal to 0.57722.

Median: \mu - \sigma\ln(\ln 2).

Variance: \sigma^2 \pi^2 / 6.

Probability density function (p.d.f):

f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}

for x in R, the set of all real numbers.

Cumulative distribution function (c.d.f):

In the \xi = 0 (Gumbel) special case

F(x) = \exp\{-\exp[-(x - \mu) / \sigma] \}

for x in R, the set of all real numbers.

Value

A Gumbel object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a hurdle negative binomial distribution

Description

Hurdle negative binomial distributions are frequently used to model counts with overdispersion and many zero observations.

Usage

HurdleNegativeBinomial(mu, theta, pi)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a hurdle negative binomial random variable with parameters mu = \mu and theta = \theta.

Support: \{0, 1, 2, 3, ...\}

Mean:

\mu \cdot \frac{\pi}{1 - F(0; \mu, \theta)}

where F(k; \mu) is the c.d.f. of the NegativeBinomial distribution.

Variance:

m \cdot \left(1 + \frac{\mu}{\theta} + \mu - m \right)

where m is the mean above.

Probability mass function (p.m.f.): P(X = 0) = 1 - \pi and for k > 0

P(X = k) = \pi \cdot \frac{f(k; \mu, \theta)}{1 - F(0; \mu, \theta)}

where f(k; \mu, \theta) is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.): P(X \le 0) = 1 - \pi and for k > 0

P(X \le k) = 1 - \pi + \pi \cdot \frac{F(k; \mu, \theta) - F(0; \mu, \theta)}{1 - F(0; \mu, \theta)}

Moment generating function (m.g.f.):

Omitted for now.

Value

A HurdleNegativeBinomial object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a hurdle Poisson distribution

Description

Hurdle Poisson distributions are frequently used to model counts with many zero observations.

Usage

HurdlePoisson(lambda, pi)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a hurdle Poisson random variable with parameter lambda = \lambda.

Support: \{0, 1, 2, 3, ...\}

Mean:

\lambda \cdot \frac{\pi}{1 - e^{-\lambda}}

Variance: m \cdot (\lambda + 1 - m), where m is the mean above.

Probability mass function (p.m.f.): P(X = 0) = 1 - \pi and for k > 0

P(X = k) = \pi \cdot \frac{f(k; \lambda)}{1 - f(0; \lambda)}

where f(k; \lambda) is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.): P(X \le 0) = 1 - \pi and for k > 0

P(X \le k) = 1 - \pi + \pi \cdot \frac{F(k; \lambda) - F(0; \lambda)}{1 - F(0; \lambda)}

where F(k; \lambda) is the c.d.f. of the Poisson distribution.

Moment generating function (m.g.f.):

Omitted for now.

Value

A HurdlePoisson object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a HyperGeometric distribution

Description

To understand the HyperGeometric distribution, consider a set of r objects, of which m are of the type I and n are of the type II. A sample with size k (k<r) with no replacement is randomly chosen. The number of observed type I elements observed in this sample is set to be our random variable X. For example, consider that in a set of 20 car parts, there are 4 that are defective (type I). If we take a sample of size 5 from those car parts, the probability of finding 2 that are defective will be given by the HyperGeometric distribution (needs double checking).

Usage

HyperGeometric(m, n, k)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a HyperGeometric random variable with success probability p = p = m/(m+n).

Support: x \in { \{\max{(0, k-n)}, \dots, \min{(k,m)}}\}

Mean: \frac{km}{n+m} = kp

Variance: \frac{km(n)(n+m-k)}{(n+m)^2 (n+m-1)} = kp(1-p)(1 - \frac{k-1}{m+n-1})

Probability mass function (p.m.f):

P(X = x) = \frac{{m \choose x}{n \choose k-x}}{{m+n \choose k}}

Cumulative distribution function (c.d.f):

P(X \le k) \approx \Phi\Big(\frac{x - kp}{\sqrt{kp(1-p)}}\Big)

Moment generating function (m.g.f):

Not useful.

Value

A HyperGeometric object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine whether a distribution is discrete or continuous

Description

Generic functions for determining whether a certain probability distribution is discrete or continuous, respectively.

Usage

is_discrete(d, ...)

is_continuous(d, ...)

Arguments

Details

The generic function is_discrete is intended to return TRUE for every distribution whose entire support is discrete and FALSE otherwise. Analogously, is_continuous is intended to return TRUE for every distribution whose entire support is continuous and FALSE otherwise. For mixed discrete-continuous distributions both methods should return FALSE.

Methods for both generics are provided for all distribution classes set up in this package.

Value

A logical vector indicating whether the distribution(s) in d is/are discrete or continuous, respectively.

Examples

X <- Normal()
is_discrete(X)
is_continuous(X)
Y <- Binomial(size = 10, p = c(0.2, 0.5, 0.8))
is_discrete(Y)
is_continuous(Y)

Is an object a distribution?

Description

is_distribution tests if x inherits from "distribution".

Usage

is_distribution(x)

Arguments

Examples

Z <- Normal()

is_distribution(Z)
is_distribution(1L)

Compute the (log-)likelihood of a probability distribution given data

Description

Functions for computing the (log-)likelihood based on a distribution object and observed data. The log-likelihood is computed as the sum of log-density contributions and the likelihood by taking the exponential thereof.

Usage

log_likelihood(d, x, ...)

likelihood(d, x, ...)

Arguments

Value

Numeric value of the (log-)likelihood.

Examples

## distribution object
X <- Normal()
## sum of log_pdf() contributions
log_likelihood(X, c(-1, 0, 0, 0, 3))
## exp of log_likelihood()
likelihood(X, c(-1, 0, 0, 0, 3))

Create a Logistic distribution

Description

A continuous distribution on the real line. For binary outcomes the model given by P(Y = 1 | X) = F(X \beta) where F is the Logistic cdf() is called logistic regression.

Usage

Logistic(location = 0, scale = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Logistic random variable with location = \mu and scale = s.

Support: R, the set of all real numbers

Mean: \mu

Variance: s^2 \pi^2 / 3

Probability density function (p.d.f):

f(x) = \frac{e^{-(\frac{x - \mu}{s})}}{s [1 + \exp(-(\frac{x - \mu}{s})) ]^2}

Cumulative distribution function (c.d.f):

F(t) = \frac{1}{1 + e^{-(\frac{t - \mu}{s})}}

Moment generating function (m.g.f):

E(e^{tX}) = e^{\mu t} \beta(1 - st, 1 + st)

where \beta(x, y) is the Beta function.

Value

A Logistic object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a LogNormal distribution

Description

A random variable created by exponentiating a Normal() distribution. Taking the log of LogNormal data returns in Normal() data.

Usage

LogNormal(log_mu = 0, log_sigma = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a LogNormal random variable with success probability p = p.

Support: R^+

Mean: \exp(\mu + \sigma^2/2)

Variance: [\exp(\sigma^2)-1]\exp(2\mu+\sigma^2)

Probability density function (p.d.f):

f(x) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp \left(-\frac{(\log x - \mu)^2}{2 \sigma^2} \right)

Cumulative distribution function (c.d.f):

F(x) = \frac{1}{2} + \frac{1}{2\sqrt{pi}}\int_{-x}^x e^{-t^2} dt

Moment generating function (m.g.f): Undefined.

Value

A LogNormal object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a Multinomial distribution

Description

The multinomial distribution is a generalization of the binomial distribution to multiple categories. It is perhaps easiest to think that we first extend a Bernoulli() distribution to include more than two categories, resulting in a Categorical() distribution. We then extend repeat the Categorical experiment several (n) times.

Usage

Multinomial(size, p)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X = (X_1, ..., X_k) be a Multinomial random variable with success probability p = p. Note that p is vector with k elements that sum to one. Assume that we repeat the Categorical experiment size = n times.

Support: Each X_i is in {0, 1, 2, ..., n}.

Mean: The mean of X_i is n p_i.

Variance: The variance of X_i is n p_i (1 - p_i). For i \neq j, the covariance of X_i and X_j is -n p_i p_j.

Probability mass function (p.m.f):

P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot ... \cdot p_k^{x_k}

Cumulative distribution function (c.d.f):

Omitted for multivariate random variables for the time being.

Moment generating function (m.g.f):

E(e^{tX}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n

Value

A Multinomial object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

Create a negative binomial distribution

Description

A generalization of the geometric distribution. It is the number of failures in a sequence of i.i.d. Bernoulli trials before a specified target number (r) of successes occurs.

Usage

NegativeBinomial(size, p = 0.5, mu = size)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a negative binomial random variable with success probability p = p.

Support: \{0, 1, 2, 3, ...\}

Mean: \frac{(1 - p) r}{p} = \mu

Variance: \frac{(1 - p) r}{p^2}

Probability mass function (p.m.f.):

f(k) = {k + r - 1 \choose k} \cdot p^r (1-p)^k

Cumulative distribution function (c.d.f.):

Omitted for now.

Moment generating function (m.g.f.):

\left(\frac{p}{1 - (1 -p) e^t}\right)^r, t < -\log (1-p)

Alternative parameterization: Sometimes, especially when used in regression models, the negative binomial distribution is parameterized by its mean \mu (as listed above) plus the size parameter r. This implies a success probability of p = r/(r + \mu). This can also be seen as a generalization of the Poisson distribution where the assumption of equidispersion (i.e., variance equal to mean) is relaxed. The negative binomial distribution is overdispersed (i.e., variance greater than mean) and its variance can also be written as \mu + 1/r \mu^2. The Poisson distribution is then obtained as r goes to infinity. Note that in this view it is natural to also allow for non-integer r parameters. The factorials in the equations above are then expressed in terms of the gamma function.

Value

A NegativeBinomial object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Create a Normal distribution

Description

The Normal distribution is ubiquitous in statistics, partially because of the central limit theorem, which states that sums of i.i.d. random variables eventually become Normal. Linear transformations of Normal random variables result in new random variables that are also Normal. If you are taking an intro stats course, you'll likely use the Normal distribution for Z-tests and in simple linear regression. Under regularity conditions, maximum likelihood estimators are asymptotically Normal. The Normal distribution is also called the gaussian distribution.

Usage

Normal(mu = 0, sigma = 1)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Normal random variable with mean mu = \mu and standard deviation sigma = \sigma.

Support: R, the set of all real numbers

Mean: \mu

Variance: \sigma^2

Probability density function (p.d.f):

f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard Normal is sometimes called the "error function". The notation \Phi(t) also stands for the c.d.f. of a standard Normal evaluated at t. Z-tables list the value of \Phi(t) for various t.

Moment generating function (m.g.f):

E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}

Value

A Normal object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability density of a probability distribution

Description

Generic function for computing probability density function (PDF) contributions based on a distribution object and observed data.

Usage

pdf(d, x, drop = TRUE, ...)

log_pdf(d, x, ...)

pmf(d, x, ...)

Arguments

Details

The generic function pdf() computes the probability density, both for continuous and discrete distributions. pmf() (for the probability mass function) is an alias that just calls pdf() internally. For computing log-density contributions (e.g., to a log-likelihood) either pdf(..., log = TRUE) can be used or the generic function log_pdf().

Value

Probabilities corresponding to the vector x.

Examples

## distribution object
X <- Normal()
## probability density
pdf(X, c(1, 2, 3, 4, 5))
pmf(X, c(1, 2, 3, 4, 5))
## log-density
pdf(X, c(1, 2, 3, 4, 5), log = TRUE)
log_pdf(X, c(1, 2, 3, 4, 5))

Evaluate the probability mass function of a Bernoulli distribution

Description

Evaluate the probability mass function of a Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Bernoulli'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Beta distribution

Description

Evaluate the probability mass function of a Beta distribution

Usage

## S3 method for class 'Beta'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Beta'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Binomial distribution

Description

Evaluate the probability mass function of a Binomial distribution

Usage

## S3 method for class 'Binomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Binomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Categorical discrete distribution

Description

Evaluate the probability mass function of a Categorical discrete distribution

Usage

## S3 method for class 'Categorical'
pdf(d, x, ...)

## S3 method for class 'Categorical'
log_pdf(d, x, ...)

Arguments

Value

A vector of probabilities, one for each element of x.

Examples

set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Evaluate the probability mass function of a Cauchy distribution

Description

Evaluate the probability mass function of a Cauchy distribution

Usage

## S3 method for class 'Cauchy'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Cauchy'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a chi square distribution

Description

Evaluate the probability mass function of a chi square distribution

Usage

## S3 method for class 'ChiSquare'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ChiSquare'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of an Erlang distribution

Description

Evaluate the probability mass function of an Erlang distribution

Usage

## S3 method for class 'Erlang'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Erlang'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability density function of an Exponential distribution

Description

Evaluate the probability density function of an Exponential distribution

Usage

## S3 method for class 'Exponential'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Exponential'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of an F distribution

Description

Evaluate the probability mass function of an F distribution

Usage

## S3 method for class 'FisherF'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'FisherF'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Frechet distribution

Description

Evaluate the probability mass function of a Frechet distribution

Usage

## S3 method for class 'Frechet'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Frechet'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Gamma distribution

Description

Evaluate the probability mass function of a Gamma distribution

Usage

## S3 method for class 'Gamma'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gamma'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Geometric distribution

Description

Please see the documentation of Geometric() for some properties of the Geometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Geometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Geometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Geometric distribution: cdf.Geometric(), quantile.Geometric(), random.Geometric()

Examples

set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a GEV distribution

Description

Evaluate the probability mass function of a GEV distribution

Usage

## S3 method for class 'GEV'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GEV'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a GP distribution

Description

Evaluate the probability mass function of a GP distribution

Usage

## S3 method for class 'GP'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GP'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Gumbel distribution

Description

Evaluate the probability mass function of a Gumbel distribution

Usage

## S3 method for class 'Gumbel'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gumbel'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a hurdle negative binomial distribution

Description

Evaluate the probability mass function of a hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdleNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a hurdle Poisson distribution

Description

Evaluate the probability mass function of a hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdlePoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a HyperGeometric distribution

Description

Please see the documentation of HyperGeometric() for some properties of the HyperGeometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'HyperGeometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HyperGeometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other HyperGeometric distribution: cdf.HyperGeometric(), quantile.HyperGeometric(), random.HyperGeometric()

Examples

set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a Logistic distribution

Description

Please see the documentation of Logistic() for some properties of the Logistic distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Logistic'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Logistic'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Logistic distribution: cdf.Logistic(), quantile.Logistic(), random.Logistic()

Examples

set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a LogNormal distribution

Description

Please see the documentation of LogNormal() for some properties of the LogNormal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'LogNormal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'LogNormal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other LogNormal distribution: cdf.LogNormal(), fit_mle.LogNormal(), quantile.LogNormal(), random.LogNormal()

Examples

set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a Multinomial distribution

Description

Please see the documentation of Multinomial() for some properties of the Multinomial distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Multinomial'
pdf(d, x, ...)

## S3 method for class 'Multinomial'
log_pdf(d, x, ...)

Arguments

Value

A vector of probabilities, one for each element of x.

See Also

Other Multinomial distribution: random.Multinomial()

Examples

set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

Evaluate the probability mass function of a NegativeBinomial distribution

Description

Evaluate the probability mass function of a NegativeBinomial distribution

Usage

## S3 method for class 'NegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'NegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other NegativeBinomial distribution: cdf.NegativeBinomial(), quantile.NegativeBinomial(), random.NegativeBinomial()

Examples

set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Evaluate the probability mass function of a Normal distribution

Description

Please see the documentation of Normal() for some properties of the Normal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Normal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Normal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Normal distribution: cdf.Normal(), fit_mle.Normal(), quantile.Normal()

Examples

set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Poisson distribution

Description

Evaluate the probability mass function of a Poisson distribution

Usage

## S3 method for class 'Poisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Poisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a PoissonBinomial distribution

Description

Evaluate the probability mass function of a PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, log = FALSE, verbose = TRUE, ...)

## S3 method for class 'PoissonBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Evaluate the probability mass function of an RevWeibull distribution

Description

Evaluate the probability mass function of an RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'RevWeibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a StudentsT distribution

Description

Please see the documentation of StudentsT() for some properties of the StudentsT distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'StudentsT'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'StudentsT'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other StudentsT distribution: cdf.StudentsT(), quantile.StudentsT(), random.StudentsT()

Examples

set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Evaluate the probability mass function of a continuous Uniform distribution

Description

Evaluate the probability mass function of a continuous Uniform distribution

Usage

## S3 method for class 'Uniform'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Uniform'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Weibull distribution

Description

Please see the documentation of Weibull() for some properties of the Weibull distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Weibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Weibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

See Also

Other Weibull distribution: cdf.Weibull(), quantile.Weibull(), random.Weibull()

Examples

set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a zero-inflated negative binomial distribution

Description

Evaluate the probability mass function of a zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZINegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-inflated Poisson distribution

Description

Evaluate the probability mass function of a zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZIPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-truncated negative binomial distribution

Description

Evaluate the probability mass function of a zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-truncated Poisson distribution

Description

Evaluate the probability mass function of a zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Plot the CDF of a distribution

Description

A function to easily plot the CDF of a distribution using ggplot2. Requires ggplot2 to be loaded.

Usage

plot_cdf(d, limits = NULL, p = 0.001, plot_theme = NULL)

Arguments

Examples

N1 <- Normal()
plot_cdf(N1)

N2 <- Normal(0, c(1, 2))
plot_cdf(N2)

B1 <- Binomial(10, 0.2)
plot_cdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_cdf(B2)

Plot the PDF of a distribution

Description

A function to easily plot the PDF of a distribution using ggplot2. Requires ggplot2 to be loaded.

Usage

plot_pdf(d, limits = NULL, p = 0.001, plot_theme = NULL)

Arguments

Examples

N1 <- Normal()
plot_pdf(N1)

N2 <- Normal(0, c(1, 2))
plot_pdf(N2)

B1 <- Binomial(10, 0.2)
plot_pdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_pdf(B2)

Plot the p.m.f, p.d.f or c.d.f. of a univariate distribution

Description

Plot method for an object inheriting from class "distribution". By default the probability density function (p.d.f.), for a continuous variable, or the probability mass function (p.m.f.), for a discrete variable, is plotted. The cumulative distribution function (c.d.f.) will be plotted if cdf = TRUE. Multiple functions are included in the plot if any of the parameter vectors in x has length greater than 1. See the argument all.

Usage

## S3 method for class 'distribution'
plot(
  x,
  cdf = FALSE,
  p = c(0.1, 99.9),
  len = 1000,
  all = FALSE,
  legend_args = list(),
  ...
)

Arguments

Details

If xlim is passed in ... then this determines the range of values of the variable to be plotted on the horizontal axis. If x is a discrete distribution object then the values for which the p.m.f. or c.d.f. is plotted is the smallest set of consecutive integers that contains both components of xlim. Otherwise, xlim is used directly.

If xlim is not passed in ... then the range of values spans the support of the distribution, with the following proviso: if the lower (upper) endpoint of the distribution is -Inf (Inf) then the lower (upper) limit of the plotting range is set to the p[1]\

If the name of x is a single upper case letter then that name is used to labels the axes of the plot. Otherwise, x and P(X = x) or f(x) are used.

A legend is included only if at least one of the parameter vectors in x has length greater than 1.

Plots of c.d.f.s are produced using calls to approxfun and plot.ecdf.

Value

An object with the same class as x, in which the parameter vectors have been expanded to contain a parameter combination for each function plotted.

Examples

B <- Binomial(20, 0.7)
plot(B)
plot(B, cdf = TRUE)

B2 <- Binomial(20, c(0.1, 0.5, 0.9))
plot(B2, legend_args = list(x = "top"))
x <- plot(B2, cdf = TRUE)
x$size
x$p

X <- Poisson(2)
plot(X)
plot(X, cdf = TRUE)

G <- Gamma(c(1, 3), 1:2)
plot(G)
plot(G, all = TRUE)
plot(G, cdf = TRUE)

C <- Cauchy()
plot(C, p = c(1, 99), len = 10000)
plot(C, cdf = TRUE, p = c(1, 99))

Create a Poisson distribution

Description

Poisson distributions are frequently used to model counts.

Usage

Poisson(lambda)

Arguments

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Poisson random variable with parameter lambda = \lambda.

Support: \{0, 1, 2, 3, ...\}

Mean: \lambda