Title: Parameter Estimation
Version: 0.8.5
Description: Implements estimation methods for parameters of common distribution families. The common d, p, q, r function family for each distribution is enriched with the ll, e, and v counterparts, computing the log-likelihood, performing estimation, and calculating the asymptotic variance - covariance matrix, respectively. Parameter estimation is performed analytically whenever possible.
License: GPL (≥ 3)
URL: https://thechibo.github.io/estimators/
BugReports: https://github.com/thechibo/estimators/issues
Depends: R (≥ 4.0.0)
Imports: extraDistr, ggh4x, ggplot2, grDevices, Matrix, methods, progress, stats, utils
Suggests: covr, knitr, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder: knitr
Config/testthat/edition: 3
Encoding: UTF-8
Language: en-US
RoxygenNote: 7.2.3
NeedsCompilation: no
Packaged: 2024-05-08 13:59:18 UTC; Chibo
Author: Ioannis Oikonomidis ORCID iD [aut, cre], Samis Trevezas ORCID iD [aut, ths]
Maintainer: Ioannis Oikonomidis <goikon@math.uoa.gr>
Repository: CRAN
Date/Publication: 2024-05-16 15:00:10 UTC

estimators: Parameter Estimation

Description

logo

Implements estimation methods for parameters of common distribution families. The common d, p, q, r function family for each distribution is enriched with the ll, e, and v counterparts, computing the log-likelihood, performing estimation, and calculating the asymptotic variance - covariance matrix, respectively. Parameter estimation is performed analytically whenever possible.

Author(s)

Maintainer: Ioannis Oikonomidis goikon@math.uoa.gr (ORCID)

Authors:

See Also

Useful links:

Bernoulli Distribution

Description

Bernoulli Distribution

Usage

Bern(prob = 0.5)

## S4 method for signature 'Bern'
d(x)

## S4 method for signature 'Bern'
p(x)

## S4 method for signature 'Bern'
qn(x)

## S4 method for signature 'Bern'
r(x)

## S4 method for signature 'Bern'
mean(x)

## S4 method for signature 'Bern'
median(x)

## S4 method for signature 'Bern'
mode(x)

## S4 method for signature 'Bern'
var(x)

## S4 method for signature 'Bern'
sd(x)

## S4 method for signature 'Bern'
skew(x)

## S4 method for signature 'Bern'
kurt(x)

## S4 method for signature 'Bern'
entro(x)

## S4 method for signature 'Bern'
finf(x)

## S4 method for signature 'numeric,numeric,Bern'
ll(x, prm, distr)

## S4 method for signature 'numeric,Bern'
mle(x, distr)

## S4 method for signature 'numeric,Bern'
me(x, distr)

## S4 method for signature 'Bern'
avar_mle(distr)

## S4 method for signature 'Bern'
avar_me(distr)

Arguments

x

an object of class Bern. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm, prob

numeric. The distribution parameter.

distr

an object of class Bern.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

See Also

dpqr, moments

Beta Distribution

Description

Beta Distribution

Usage

Beta(shape1 = 1, shape2 = 1, ncp = 0)

## S4 method for signature 'Beta'
d(x)

## S4 method for signature 'Beta'
p(x)

## S4 method for signature 'Beta'
qn(x)

## S4 method for signature 'Beta'
r(x)

## S4 method for signature 'Beta'
mean(x)

## S4 method for signature 'Beta'
median(x)

## S4 method for signature 'Beta'
mode(x)

## S4 method for signature 'Beta'
var(x)

## S4 method for signature 'Beta'
sd(x)

## S4 method for signature 'Beta'
skew(x)

## S4 method for signature 'Beta'
kurt(x)

## S4 method for signature 'Beta'
entro(x)

## S4 method for signature 'Beta'
finf(x)

llbeta(x, shape1, shape2)

## S4 method for signature 'numeric,numeric,Beta'
ll(x, prm, distr)

## S4 method for signature 'numeric,Beta'
mle(x, distr, par0 = "same", method = "L-BFGS-B", lower = 1e-05, upper = Inf)

## S4 method for signature 'numeric,Beta'
me(x, distr)

## S4 method for signature 'numeric,Beta'
same(x, distr)

vbeta(shape1, shape2, type = "mle")

## S4 method for signature 'Beta'
avar_mle(distr)

## S4 method for signature 'Beta'
avar_me(distr)

## S4 method for signature 'Beta'
avar_same(distr)

Arguments

shape1, shape2, ncp

numeric. The distribution parameters.

x

an object of class Beta. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Beta.

par0, method, lower, upper

arguments passed to optim.

type

character, case ignored. The estimator type (mle, me, or same).

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Binomial Distribution

Description

Binomial Distribution

Usage

Binom(size = 1, prob = 0.5)

## S4 method for signature 'Binom'
d(x)

## S4 method for signature 'Binom'
p(x)

## S4 method for signature 'Binom'
qn(x)

## S4 method for signature 'Binom'
r(x)

## S4 method for signature 'Binom'
mean(x)

## S4 method for signature 'Binom'
var(x)

## S4 method for signature 'Binom'
sd(x)

## S4 method for signature 'Binom'
skew(x)

## S4 method for signature 'Binom'
kurt(x)

## S4 method for signature 'Binom'
finf(x)

## S4 method for signature 'numeric,numeric,Binom'
ll(x, prm, distr)

## S4 method for signature 'numeric,Binom'
mle(x, distr)

## S4 method for signature 'numeric,Binom'
me(x, distr)

## S4 method for signature 'Binom'
avar_mle(distr)

## S4 method for signature 'Binom'
avar_me(distr)

Arguments

size, prob

numeric. The distribution parameters.

x

an object of class Binom. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Binom.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Categorical Distribution

Description

Categorical Distribution

Usage

Cat(prob = c(0.5, 0.5))

## S4 method for signature 'Cat'
d(x)

## S4 method for signature 'Cat'
r(x)

## S4 method for signature 'Cat'
mean(x)

## S4 method for signature 'Cat'
var(x)

## S4 method for signature 'Cat'
finf(x)

## S4 method for signature 'numeric,numeric,Cat'
ll(x, prm, distr)

## S4 method for signature 'numeric,Cat'
mle(x, distr)

## S4 method for signature 'numeric,Cat'
me(x, distr)

## S4 method for signature 'Cat'
avar_mle(distr)

## S4 method for signature 'Cat'
avar_me(distr)

Arguments

prob

numeric. The distribution parameters.

x

an object of class Cat. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Cat.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Cauchy Distribution

Description

Cauchy Distribution

Usage

Cauchy(location = 1, scale = 1)

## S4 method for signature 'Cauchy'
d(x)

## S4 method for signature 'Cauchy'
p(x)

## S4 method for signature 'Cauchy'
qn(x)

## S4 method for signature 'Cauchy'
r(x)

## S4 method for signature 'Cauchy'
median(x)

## S4 method for signature 'Cauchy'
mode(x)

## S4 method for signature 'Cauchy'
entro(x)

## S4 method for signature 'Cauchy'
finf(x)

## S4 method for signature 'numeric,numeric,Cauchy'
ll(x, prm, distr)

Arguments

location, scale

numeric. The distribution parameters.

x

an object of class Cauchy. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Cauchy.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Chi-Square Distribution

Description

Chi-Square Distribution

Usage

Chisq(df = 1, ncp = 0)

## S4 method for signature 'Chisq'
d(x)

## S4 method for signature 'Chisq'
p(x)

## S4 method for signature 'Chisq'
qn(x)

## S4 method for signature 'Chisq'
r(x)

## S4 method for signature 'Chisq'
mean(x)

## S4 method for signature 'Chisq'
var(x)

## S4 method for signature 'Chisq'
sd(x)

## S4 method for signature 'Chisq'
skew(x)

## S4 method for signature 'Chisq'
kurt(x)

Arguments

df, ncp

numeric. The distribution parameters.

x

an object of class Chisq. If the function also has a distr argument, x is a numeric vector, a sample of observations.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Dirichlet Distribution

Description

Dirichlet Distribution

Usage

Dir(alpha = c(1, 1))

## S4 method for signature 'Dir'
d(x)

## S4 method for signature 'Dir'
r(x)

## S4 method for signature 'Dir'
mean(x)

## S4 method for signature 'Dir'
mode(x)

## S4 method for signature 'Dir'
var(x)

## S4 method for signature 'Dir'
entro(x)

## S4 method for signature 'Dir'
finf(x)

## S4 method for signature 'matrix,numeric,Dir'
ll(x, prm, distr)

## S4 method for signature 'matrix,Dir'
mle(x, distr, par0 = "same", method = "L-BFGS-B", lower = 1e-05, upper = Inf)

## S4 method for signature 'matrix,Dir'
me(x, distr)

## S4 method for signature 'matrix,Dir'
same(x, distr)

## S4 method for signature 'Dir'
avar_mle(distr)

## S4 method for signature 'Dir'
avar_me(distr)

## S4 method for signature 'Dir'
avar_same(distr)

Arguments

alpha

numeric. The distribution parameters.

x

an object of class Dir. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Dir.

par0, method, lower, upper

arguments passed to optim.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Distribution S4 Classes

Description

A collection of classes that provide a flexible and structured way to work with probability distributions.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Exponential Distribution

Description

Exponential Distribution

Usage

Exp(rate = 1)

## S4 method for signature 'Exp'
d(x)

## S4 method for signature 'Exp'
p(x)

## S4 method for signature 'Exp'
qn(x)

## S4 method for signature 'Exp'
r(x)

## S4 method for signature 'Exp'
mean(x)

## S4 method for signature 'Exp'
median(x)

## S4 method for signature 'Exp'
mode(x)

## S4 method for signature 'Exp'
var(x)

## S4 method for signature 'Exp'
sd(x)

## S4 method for signature 'Exp'
skew(x)

## S4 method for signature 'Exp'
kurt(x)

## S4 method for signature 'Exp'
entro(x)

## S4 method for signature 'Exp'
finf(x)

## S4 method for signature 'numeric,numeric,Exp'
ll(x, prm, distr)

## S4 method for signature 'numeric,Exp'
mle(x, distr)

## S4 method for signature 'numeric,Exp'
me(x, distr)

## S4 method for signature 'Exp'
avar_mle(distr)

## S4 method for signature 'Exp'
avar_me(distr)

Arguments

rate

numeric. The distribution parameters.

x

an object of class Exp. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Exp.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Fisher Distribution

Description

Fisher Distribution

Usage

Fisher(df1 = 1, df2 = 1, ncp = 0)

## S4 method for signature 'Fisher'
d(x)

## S4 method for signature 'Fisher'
p(x)

## S4 method for signature 'Fisher'
qn(x)

## S4 method for signature 'Fisher'
r(x)

## S4 method for signature 'Fisher'
mean(x)

## S4 method for signature 'Fisher'
mode(x)

## S4 method for signature 'Fisher'
var(x)

## S4 method for signature 'Fisher'
sd(x)

## S4 method for signature 'Fisher'
skew(x)

Arguments

df1, df2, ncp

numeric. The distribution parameters.

x

an object of class Fisher. If the function also has a distr argument, x is a numeric vector, a sample of observations.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Gamma Distribution

Description

Gamma Distribution

Usage

Gam(shape = 1, scale = 1)

## S4 method for signature 'Gam'
d(x)

## S4 method for signature 'Gam'
p(x)

## S4 method for signature 'Gam'
qn(x)

## S4 method for signature 'Gam'
r(x)

## S4 method for signature 'Gam'
mean(x)

## S4 method for signature 'Gam'
var(x)

## S4 method for signature 'Gam'
sd(x)

## S4 method for signature 'Gam'
skew(x)

## S4 method for signature 'Gam'
kurt(x)

## S4 method for signature 'Gam'
entro(x)

## S4 method for signature 'Gam'
finf(x)

## S4 method for signature 'numeric,numeric,Gam'
ll(x, prm, distr)

## S4 method for signature 'numeric,Gam'
mle(x, distr, par0 = "same", method = "L-BFGS-B", lower = 1e-05, upper = Inf)

## S4 method for signature 'numeric,Gam'
me(x, distr)

## S4 method for signature 'numeric,Gam'
same(x, distr)

## S4 method for signature 'Gam'
avar_mle(distr)

## S4 method for signature 'Gam'
avar_me(distr)

## S4 method for signature 'Gam'
avar_same(distr)

Arguments

shape, scale

numeric. The distribution parameters.

x

an object of class Gam. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Gam.

par0, method, lower, upper

arguments passed to optim.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Geometric Distribution

Description

Geometric Distribution

Usage

Geom(prob = 0.5)

## S4 method for signature 'Geom'
d(x)

## S4 method for signature 'Geom'
p(x)

## S4 method for signature 'Geom'
qn(x)

## S4 method for signature 'Geom'
r(x)

## S4 method for signature 'Geom'
mean(x)

## S4 method for signature 'Geom'
mode(x)

## S4 method for signature 'Geom'
var(x)

## S4 method for signature 'Geom'
sd(x)

## S4 method for signature 'Geom'
skew(x)

## S4 method for signature 'Geom'
kurt(x)

## S4 method for signature 'Geom'
entro(x)

## S4 method for signature 'Geom'
finf(x)

## S4 method for signature 'numeric,numeric,Geom'
ll(x, prm, distr)

## S4 method for signature 'numeric,Geom'
mle(x, distr)

## S4 method for signature 'numeric,Geom'
me(x, distr)

## S4 method for signature 'Geom'
avar_mle(distr)

## S4 method for signature 'Geom'
avar_me(distr)

Arguments

prob

numeric. The distribution parameters.

x

an object of class Geom. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Geom.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Laplace Distribution

Description

Laplace Distribution

Usage

Laplace(mu = 0, sigma = 1)

## S4 method for signature 'Laplace'
d(x)

## S4 method for signature 'Laplace'
p(x)

## S4 method for signature 'Laplace'
qn(x)

## S4 method for signature 'Laplace'
r(x)

## S4 method for signature 'Laplace'
mean(x)

## S4 method for signature 'Laplace'
median(x)

## S4 method for signature 'Laplace'
mode(x)

## S4 method for signature 'Laplace'
var(x)

## S4 method for signature 'Laplace'
sd(x)

## S4 method for signature 'Laplace'
skew(x)

## S4 method for signature 'Laplace'
kurt(x)

## S4 method for signature 'Laplace'
entro(x)

## S4 method for signature 'Laplace'
finf(x)

## S4 method for signature 'numeric,numeric,Laplace'
ll(x, prm, distr)

## S4 method for signature 'numeric,Laplace'
mle(x, distr)

## S4 method for signature 'numeric,Laplace'
me(x, distr)

## S4 method for signature 'Laplace'
avar_mle(distr)

## S4 method for signature 'Laplace'
avar_me(distr)

Arguments

mu, sigma

numeric. The distribution parameters.

x

an object of class Laplace. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Laplace.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Lnorm Distribution

Description

Lnorm Distribution

Usage

Lnorm(meanlog = 0, sdlog = 1)

## S4 method for signature 'Lnorm'
d(x)

## S4 method for signature 'Lnorm'
p(x)

## S4 method for signature 'Lnorm'
qn(x)

## S4 method for signature 'Lnorm'
r(x)

## S4 method for signature 'Lnorm'
mean(x)

## S4 method for signature 'Lnorm'
median(x)

## S4 method for signature 'Lnorm'
mode(x)

## S4 method for signature 'Lnorm'
var(x)

## S4 method for signature 'Lnorm'
sd(x)

## S4 method for signature 'Lnorm'
skew(x)

## S4 method for signature 'Lnorm'
kurt(x)

## S4 method for signature 'Lnorm'
entro(x)

## S4 method for signature 'Lnorm'
finf(x)

## S4 method for signature 'numeric,numeric,Lnorm'
ll(x, prm, distr)

## S4 method for signature 'numeric,Lnorm'
mle(x, distr)

## S4 method for signature 'numeric,Lnorm'
me(x, distr)

## S4 method for signature 'Lnorm'
avar_mle(distr)

## S4 method for signature 'Lnorm'
avar_me(distr)

Arguments

meanlog, sdlog

numeric. The distribution parameters.

x

an object of class Lnorm. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Lnorm.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Multinomial Distribution

Description

Multinomial Distribution

Usage

Multinom(size = 1, prob = c(0.5, 0.5))

## S4 method for signature 'Multinom'
d(x)

## S4 method for signature 'Multinom'
r(x)

## S4 method for signature 'Multinom'
mean(x)

## S4 method for signature 'Multinom'
var(x)

## S4 method for signature 'Multinom'
finf(x)

## S4 method for signature 'matrix,numeric,Multinom'
ll(x, prm, distr)

## S4 method for signature 'matrix,Multinom'
mle(x, distr)

## S4 method for signature 'matrix,Multinom'
me(x, distr)

## S4 method for signature 'Multinom'
avar_mle(distr)

## S4 method for signature 'Multinom'
avar_me(distr)

Arguments

size, prob

numeric. The distribution parameters.

x

an object of class Multinom. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Multinom.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Negative Binomial Distribution

Description

Negative Binomial Distribution

Usage

Nbinom(size = 1, prob = 0.5)

## S4 method for signature 'Nbinom'
d(x)

## S4 method for signature 'Nbinom'
p(x)

## S4 method for signature 'Nbinom'
qn(x)

## S4 method for signature 'Nbinom'
r(x)

## S4 method for signature 'Nbinom'
mean(x)

## S4 method for signature 'Nbinom'
mode(x)

## S4 method for signature 'Nbinom'
var(x)

## S4 method for signature 'Nbinom'
sd(x)

## S4 method for signature 'Nbinom'
skew(x)

## S4 method for signature 'Nbinom'
kurt(x)

## S4 method for signature 'Nbinom'
finf(x)

## S4 method for signature 'numeric,numeric,Nbinom'
ll(x, prm, distr)

## S4 method for signature 'numeric,Nbinom'
mle(x, distr)

## S4 method for signature 'numeric,Nbinom'
me(x, distr)

## S4 method for signature 'Nbinom'
avar_mle(distr)

## S4 method for signature 'Nbinom'
avar_me(distr)

Arguments

size, prob

numeric. The distribution parameters.

x

an object of class Nbinom. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Nbinom.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Normal Distribution

Description

Normal Distribution

Usage

Norm(mean = 0, sd = 1)

## S4 method for signature 'Norm'
d(x)

## S4 method for signature 'Norm'
p(x)

## S4 method for signature 'Norm'
qn(x)

## S4 method for signature 'Norm'
r(x)

## S4 method for signature 'Norm'
mean(x)

## S4 method for signature 'Norm'
median(x)

## S4 method for signature 'Norm'
mode(x)

## S4 method for signature 'Norm'
var(x)

## S4 method for signature 'Norm'
sd(x)

## S4 method for signature 'Norm'
skew(x)

## S4 method for signature 'Norm'
kurt(x)

## S4 method for signature 'Norm'
entro(x)

## S4 method for signature 'Norm'
finf(x)

## S4 method for signature 'numeric,numeric,Norm'
ll(x, prm, distr)

## S4 method for signature 'numeric,Norm'
mle(x, distr)

## S4 method for signature 'numeric,Norm'
me(x, distr)

## S4 method for signature 'Norm'
avar_mle(distr)

## S4 method for signature 'Norm'
avar_me(distr)

Arguments

mean, sd

numeric. The distribution parameters.

x

an object of class Norm. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Norm.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Poisson Distribution

Description

Poisson Distribution

Usage

Pois(lambda = 1)

## S4 method for signature 'Pois'
d(x)

## S4 method for signature 'Pois'
p(x)

## S4 method for signature 'Pois'
qn(x)

## S4 method for signature 'Pois'
r(x)

## S4 method for signature 'Pois'
mean(x)

## S4 method for signature 'Pois'
var(x)

## S4 method for signature 'Pois'
sd(x)

## S4 method for signature 'Pois'
skew(x)

## S4 method for signature 'Pois'
kurt(x)

## S4 method for signature 'Pois'
finf(x)

## S4 method for signature 'numeric,numeric,Pois'
ll(x, prm, distr)

## S4 method for signature 'numeric,Pois'
mle(x, distr)

## S4 method for signature 'numeric,Pois'
me(x, distr)

## S4 method for signature 'Pois'
avar_mle(distr)

## S4 method for signature 'Pois'
avar_me(distr)

Arguments

lambda

numeric. The distribution parameters.

x

an object of class Pois. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Pois.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Student Distribution

Description

Student Distribution

Usage

Stud(df = 1, ncp = 0)

## S4 method for signature 'Stud'
d(x)

## S4 method for signature 'Stud'
p(x)

## S4 method for signature 'Stud'
qn(x)

## S4 method for signature 'Stud'
r(x)

## S4 method for signature 'Stud'
mean(x)

## S4 method for signature 'Stud'
median(x)

## S4 method for signature 'Stud'
mode(x)

## S4 method for signature 'Stud'
var(x)

## S4 method for signature 'Stud'
sd(x)

## S4 method for signature 'Stud'
skew(x)

## S4 method for signature 'Stud'
kurt(x)

Arguments

df, ncp

numeric. The distribution parameters.

x

an object of class Stud. If the function also has a distr argument, x is a numeric vector, a sample of observations.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Uniform Distribution

Description

Uniform Distribution

Usage

Unif(min = 0, max = 1)

## S4 method for signature 'Unif'
d(x)

## S4 method for signature 'Unif'
p(x)

## S4 method for signature 'Unif'
qn(x)

## S4 method for signature 'Unif'
r(x)

## S4 method for signature 'Unif'
mean(x)

## S4 method for signature 'Unif'
var(x)

## S4 method for signature 'Unif'
sd(x)

## S4 method for signature 'Unif'
skew(x)

## S4 method for signature 'Unif'
kurt(x)

## S4 method for signature 'Unif'
entro(x)

## S4 method for signature 'numeric,numeric,Unif'
ll(x, prm, distr)

## S4 method for signature 'numeric,Unif'
mle(x, distr)

## S4 method for signature 'numeric,Unif'
me(x, distr)

Arguments

min, max

numeric. The distribution parameters.

x

an object of class Unif. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Unif.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Weibull Distribution

Description

Weibull Distribution

Usage

Weib(shape = 1, scale = 1)

## S4 method for signature 'Weib'
d(x)

## S4 method for signature 'Weib'
p(x)

## S4 method for signature 'Weib'
qn(x)

## S4 method for signature 'Weib'
r(x)

## S4 method for signature 'Weib'
mean(x)

## S4 method for signature 'Weib'
median(x)

## S4 method for signature 'Weib'
mode(x)

## S4 method for signature 'Weib'
var(x)

## S4 method for signature 'Weib'
sd(x)

## S4 method for signature 'Weib'
skew(x)

## S4 method for signature 'Weib'
kurt(x)

## S4 method for signature 'Weib'
entro(x)

## S4 method for signature 'numeric,numeric,Weib'
ll(x, prm, distr)

## S4 method for signature 'numeric,Weib'
mle(x, distr, par0 = "same", method = "L-BFGS-B", lower = 1e-05, upper = Inf)

## S4 method for signature 'numeric,Weib'
me(x, distr)

## S4 method for signature 'Weib'
avar_mle(distr)

## S4 method for signature 'Weib'
avar_me(distr)

Arguments

shape, scale

numeric. The distribution parameters.

x

an object of class Weib. If the function also has a distr argument, x is a numeric vector, a sample of observations.

prm

numeric. A vector including the distribution parameters.

distr

an object of class Weib.

par0, method, lower, upper

arguments passed to optim.

Value

The dpqr family of functions return the evaluated density, cumulative probability, quantile, and random sample, respectively. The moments family of functions return the appropriate theoretical moment, as calculated by the distribution true parameters. The ll function returns the evaluated log-likelihood, given a sample and the theoretical parameters. The estim family of functions return the estimated parameters of the distribution, given a sample. The avar family of functions return the asymptotic variance or variance - covariance matrix (if there are two or more parameters) of the corresponding estimation method. Calculus performed on Distribution objects returns a Distribution object of the appropriate class and with the appropriate parameters.

Asymptotic Variance

Description

Calculates the asymptotic variance (or variance - covariance matrix in the multidimensional case) of an estimator, given a specified family of distributions and the true parameter values.

Usage

avar(distr, type, ...)

vbern(prob, type = "mle")

vbinom(size, prob, type = "mle")

vcat(prob, type = "mle")

vdirichlet(alpha, type = "mle")

vexp(rate, type = "mle")

vgamma(shape, scale, type = "mle")

vgeom(prob, type = "mle")

vlaplace(mu, sigma, type = "mle")

vmultinom(size, prob, type = "mle")

vnbinom(size, prob, type = "mle")

vnorm(mean, sd, type = "mle")

vpois(lambda, type = "mle")

vweib(shape, scale, type = "mle")

Arguments

distr

A subclass of Distribution. The distribution family assumed.

type

character, case ignored. The estimator type (mle, me, or same).

...

extra arguments.

alpha, mu, sigma, size, prob, shape, rate, scale, mean, sd, lambda

numeric. Distribution parameters.

Value

A named matrix. The asymptotic covariance matrix of the estimator.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

avar_mle, avar_me, avar_same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

ME Asymptotic Variance

Description

Calculates the asymptotic variance (or variance - covariance matrix in the multidimensional case) of the ME, given a specified family of distributions and the true parameter values.

Usage

avar_me(distr, ...)

Arguments

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

A named matrix. The asymptotic covariance matrix of the estimator.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

avar, avar_mle, avar_same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

MLE Asymptotic Variance

Description

Calculates the asymptotic variance (or variance - covariance matrix in the multidimensional case) of the MLE, given a specified family of distributions and the true parameter values.

Usage

avar_mle(distr, ...)

Arguments

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

A named matrix. The asymptotic covariance matrix of the estimator.

See Also

avar, avar_me, avar_same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

SAME Asymptotic Variance

Description

Calculates the asymptotic variance (or variance - covariance matrix in the multidimensional case) of the SAME, given a specified family of distributions and the true parameter values.

Usage

avar_same(distr, ...)

Arguments

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

A named matrix. The asymptotic covariance matrix of the estimator.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

avar, avar_mle, avar_me

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

Distribution Calculus

Description

Distribution Calculus

Usage

## S4 method for signature 'Norm,Norm'
e1 + e2

## S4 method for signature 'numeric,Norm'
e1 + e2

## S4 method for signature 'Norm,numeric'
e1 + e2

## S4 method for signature 'Norm,Norm'
e1 - e2

## S4 method for signature 'numeric,Norm'
e1 - e2

## S4 method for signature 'Norm,numeric'
e1 - e2

## S4 method for signature 'numeric,Norm'
e1 * e2

## S4 method for signature 'Norm,numeric'
e1 * e2

## S4 method for signature 'Norm,numeric'
e1 / e2

## S4 method for signature 'Norm,logical'
sum(x, ..., na.rm = FALSE)

## S4 method for signature 'Norm'
exp(x)

Arguments

x, e1, e2

objects of subclass Distribution.

...

extra arguments.

na.rm

logical. Should missing values be removed?

Value

All calculations return Distribution objects (specifically, objects of a class that is a subclass of Distribution), accordingly to the property at hand.

Examples

# -----------------------------------------------------
# Distribution Calculus Example
# -----------------------------------------------------

library(estimators)

# Normal location - scale transformation
x <- Norm(mean = 2, sd = 3)
y <- 3 * x + 1 # Norm(mean = 7, sd = 9)

# Addition of two independent Normal random variables
x1 <- Norm(mean = 1, sd = 3)
x2 <- Norm(mean = 2, sd = 4)
x3 <- x1 + x2 # Norm(mean = 3, sd = 5)

The d p q r Functions

Description

Four generic functions that take a distribution object (e.g. Bern) and return the density, cumulative probability, quantile, and random generator functions, respectively.

Usage

d(x, ...)

p(x, ...)

qn(x, ...)

r(x, ...)

Arguments

x

an object of subclass Distribution.

...

extra arguments.

Value

The d p q r functions return the density, cumulative probability, quantile, and random generator functions, respectively.

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

library(estimators)

# Create the distribution
x <- Beta(3, 5)

# Density function
df <- d(x)
df(c(0.3, 0.8, 0.5))

# Probability function
pf <- p(x)
pf(c(0.3, 0.8, 0.5))

# Density function
qf <- qn(x)
qf(c(0.3, 0.8, 0.5))

# Random Generator function
rf <- r(x)
rf(5)

Parameter Estimation

Description

Estimates the parameters of a random sample according to a specified family of distributions.

Usage

estim(x, distr, type = "mle", ...)

ebern(x, type = "mle", ...)

ebeta(x, type = "mle", ...)

ebinom(x, type = "mle", ...)

ecat(x, type = "mle", ...)

edirichlet(x, type = "mle", ...)

eexp(x, type = "mle", ...)

egamma(x, type = "mle", ...)

egeom(x, type = "mle", ...)

elaplace(x, type = "mle", ...)

elnorm(x, type = "mle", ...)

emultinom(x, type = "mle", ...)

enbinom(x, type = "mle", ...)

enorm(x, type = "mle", ...)

epois(x, type = "mle", ...)

eunif(x, type = "mle", ...)

eweib(x, type = "mle", ...)

Arguments

x

numeric. A sample under estimation.

distr

A subclass of Distribution. The distribution family assumed.

type

character, case ignored. The estimator type (mle, me, or same).

...

extra arguments.

Value

numeric. The estimator produced by the sample.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

mle, me, same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

Polygamma Functions

Description

This set of functions revolve around the polygamma functions.

Usage

idigamma(x)

Ddigamma(x, y)

Dtrigamma(x, y)

gammap(x, p, log = FALSE)

Arguments

x, y

numeric. The points to evaluate the function.

p

integer. The p-variate Gamma function.

log

logical. Should the logarithm of the result be returned?

Value

numeric. The evaluated function.

Functions

Examples

idigamma(2)
Ddigamma(2, 3)
Dtrigamma(2, 3)
gammap(1:3, 3)

Large Sample Metrics

Description

This function performs Monte Carlo simulations to estimate the asymptotic variance - covariance matrix, characterizing the large sample behavior of an estimator. The function evaluates the metrics as a function of a single parameter, keeping the other ones constant. See Details.

Usage

large_metrics(D, prm, est = c("same", "me", "mle"), ...)

Arguments

D

A subclass of Distribution. The distribution family of interest.

prm

A list containing three elements (name, pos, val). See Details.

est

character. The estimator of interest. Can be a vector.

...

extra arguments.

Details

The distribution D is used to specify an initial distribution. The list prm contains details concerning a single parameter that is allowed to change values. The quantity of interest is evaluated as a function of this parameter.

Specifically, prm includes three elements named "name", "pos", and "val". The first two elements determine the exact parameter that changes, while the third one is a numeric vector holding the values it takes. For example, in the case of the Multivariate Gamma distribution, D <- MGamma(shape = c(1, 2), scale = 3) and prm <- list(name = "shape", pos = 2, val = seq(1, 1.5, by = 0.1)) means that the evaluation will be performed for the MGamma distributions with shape parameters ⁠(1, 1)⁠, ⁠(1, 1.1)⁠, ..., ⁠(1, 1.5)⁠ and scale 3. Notice that the initial shape parameter 2 in D is not utilized in the function.

Value

A data.frame with columns "Row", "Col", "Parameter", "Estimator", and "Value".

See Also

small_metrics, plot_small_metrics, plot_large_metrics

Examples


D <- Beta(shape1 = 1, shape2 = 2)

prm <- list(name = "shape1",
            pos = NULL,
            val = seq(0.5, 2, by = 0.5))

x <- large_metrics(D, prm,
                   est = c("mle", "me", "same"))

plot_large_metrics(x)

Log-Likelihood

Description

These functions calculate the log-likelihood of an IID sample for specific values of the distribution parameters. See Details.

Usage

ll(x, prm, distr, ...)

## S4 method for signature 'ANY,ANY,character'
ll(x, prm, distr, ...)

llbern(x, prob)

llbinom(x, size, prob)

llcat(x, prob)

llcauchy(x, location, scale)

lldirichlet(x, alpha)

llexp(x, rate)

llgamma(x, shape, scale)

llgeom(x, prob)

lllaplace(x, mu, sigma)

lllnorm(x, meanlog, sdlog)

llMultinom(x, size, prob)

llnbinom(x, size, prob)

llnorm(x, mean, sd)

llpois(x, lambda)

llunif(x, min, max)

llweib(x, shape, scale)

Arguments

x

numeric. A sample under estimation.

prm

numeric. A vector of the distribution parameters.

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

location, alpha, mu, sigma, meanlog, sdlog, min, max, size, prob, shape, rate, scale, mean, sd, lambda

numeric. Distribution parameters.

Details

The log-likelihood functions are provided in two forms: the ⁠ll<name>⁠ distribution-specific version that follows the base R conventions, and the S4 generic ll.

Value

Numeric. The value of the log-likelihood function.

Moment Estimation

Description

Calculates the ME under the assumption the sample observations are independent and identically distributed (iid) according to a specified family of distributions.

Usage

me(x, distr, ...)

## S4 method for signature 'ANY,character'
me(x, distr, ...)

Arguments

x

numeric. A sample under estimation.

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

numeric. The estimator produced by the sample.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

estim, mle, same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

Maximum Likelihood Estimation

Description

Calculates the MLE under the assumption the sample observations are independent and identically distributed (iid) according to a specified family of distributions.

Usage

mle(x, distr, ...)

## S4 method for signature 'ANY,character'
mle(x, distr, ...)

Arguments

x

numeric. A sample under estimation.

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

numeric. The estimator produced by the sample.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

estim, me, same

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

Moments - Parametric Quantities of Interest

Description

A set of functions that calculate the theoretical moments (expectation, variance, skewness, excess kurtosis) and other important parametric functions (median, mode, entropy, Fisher information) of a distribution.

Usage

moments(x)

mean(x, ...)

median(x, na.rm = FALSE, ...)

mode(x)

var(x, y = NULL, na.rm = FALSE, use)

sd(x, na.rm = FALSE)

skew(x, ...)

kurt(x, ...)

entro(x, ...)

finf(x, ...)

Arguments

x

an object of a Distribution subclass.

...

extra arguments.

y, use, na.rm

arguments in mean and var standard methods from the stats package not used here.

Details

The moments() function automatically finds the available methods for a given distribution and results all of the results in a list.

Not all functions are available for distributions; for example, the sd() is available only for univariate distributions.

Value

Numeric, either vector or matrix depending on the moment and the distribution. Function moments() returns a list of all available methods.

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

library(estimators)

# Create the distribution
x <- Beta(3, 5)

# List of all available moments
mom <- moments(x)

# Expectation
mean(x)
mom$mean

# Variance and Standard Deviation
var(x)
sd(x)

# Skewness and Excess Kurtosis
skew(x)
kurt(x)

# Entropy
entro(x)

# Fisher Information Matrix
finf(x)

Plot Large Sample Metrics

Description

This function provides an easy way to illustrate the output of large_metrics(), using the ggplot2 package. A grid of line charts is created for each element of the asymptotic variance - covariance matrix. Each estimator is plotted with a different color and linetype. The plot can be saved in pdf format.

Usage

plot_large_metrics(
  x,
  colors = NULL,
  title = NULL,
  save = FALSE,
  path = NULL,
  name = "myplot.pdf",
  width = 15,
  height = 8
)

Arguments

x

A data.frame. The result of small_metrics().

colors

character. The colors to be used in the plot.

title

character. The plot title.

save

logical. Should the plot be saved?

path

A path to the directory in which the plot will be saved.

name

character. The name of the output pdf file.

width

numeric. The plot width in inches.

height

numeric. The plot height in inches.

Value

The plot is returned invisibly in the form of a ggplot object.

See Also

small_metrics, large_metrics, plot_small_metrics

Examples


D <- Beta(shape1 = 1, shape2 = 2)

prm <- list(name = "shape1",
            pos = NULL,
            val = seq(0.5, 2, by = 0.5))

x <- small_metrics(D, prm,
                   est = c("mle", "me", "same"),
                   obs = c(20, 50),
                   sam = 1e2,
                   seed = 1)

plot_small_metrics(x)

Plot Small Sample Metrics

Description

This function provides an easy way to illustrate the output of small_metrics(), using the ggplot2 package. A grid of line charts is created for each metric and sample size. Each estimator is plotted with a different color and linetype. The plot can be saved in pdf format.

Usage

plot_small_metrics(
  x,
  colors = NULL,
  title = NULL,
  save = FALSE,
  path = NULL,
  name = "myplot.pdf",
  width = 15,
  height = 8
)

Arguments

x

A data.frame. The result of small_metrics().

colors

character. The colors to be used in the plot.

title

character. The plot title.

save

logical. Should the plot be saved?

path

A path to the directory in which the plot will be saved.

name

character. The name of the output pdf file.

width

numeric. The plot width in inches.

height

numeric. The plot height in inches.

Value

The plot is returned invisibly in the form of a ggplot object.

See Also

small_metrics, large_metrics, plot_large_metrics

Examples


D <- Beta(shape1 = 1, shape2 = 2)

prm <- list(name = "shape1",
            pos = NULL,
            val = seq(0.5, 2, by = 0.5))

x <- small_metrics(D, prm,
                   est = c("mle", "me", "same"),
                   obs = c(20, 50),
                   sam = 1e2,
                   seed = 1)

plot_small_metrics(x)

Score - Adjusted Moment Estimation

Description

Calculates the SAME under the assumption the sample observations are independent and identically distributed (iid) according to a specified family of distributions.

Usage

same(x, distr, ...)

## S4 method for signature 'ANY,character'
same(x, distr, ...)

Arguments

x

numeric. A sample under estimation.

distr

A subclass of Distribution. The distribution family assumed.

...

extra arguments.

Value

numeric. The estimator produced by the sample.

References

Ye, Z.-S. & Chen, N. (2017), Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician 71(2), 177–181.

Van der Vaart, A. W. (2000), Asymptotic statistics, Vol. 3, Cambridge university press.

Tamae, H., Irie, K. & Kubokawa, T. (2020), A score-adjusted approach to closed-form estimators for the gamma and beta distributions, Japanese Journal of Statistics and Data Science 3, 543–561.

Mathal, A. & Moschopoulos, P. (1992), A form of multivariate gamma distribution, Annals of the Institute of Statistical Mathematics 44, 97–106.

Oikonomidis, I. & Trevezas, S. (2023), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

See Also

estim, mle, me

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Simulation
set.seed(1)
shape1 <- 1
shape2 <- 2
D <- Beta(shape1, shape2)
x <- r(D)(100)

# Likelihood - The ll Functions

llbeta(x, shape1, shape2)
ll(x, c(shape1, shape2), D)
ll(x, c(shape1, shape2), "beta")

# Point Estimation - The e Functions

ebeta(x, type = "mle")
ebeta(x, type = "me")
ebeta(x, type = "same")

mle(x, D)
me(x, D)
same(x, D)

estim(x, D, type = "mle")

# Asymptotic Variance - The v Functions

vbeta(shape1, shape2, type = "mle")
vbeta(shape1, shape2, type = "me")
vbeta(shape1, shape2, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

avar(D, type = "mle")

Small Sample Metrics

Description

This function performs Monte Carlo simulations to estimate the main metrics (bias, variance, and RMSE) characterizing the small sample behavior of an estimator. The function evaluates the metrics as a function of a single parameter, keeping the other ones constant. See Details.

Usage

small_metrics(
  D,
  prm,
  est = c("same", "me", "mle"),
  obs = c(20, 50, 100),
  sam = 10000,
  seed = 1,
  ...
)

Arguments

D

A subclass of Distribution. The distribution family of interest.

prm

A list containing three elements (name, pos, val). See Details.

est

character. The estimator of interest. Can be a vector.

obs

numeric. The size of each sample. Can be a vector.

sam

numeric. The number of Monte Carlo samples used to estimate the metrics.

seed

numeric. Passed to set.seed() for reproducibility.

...

extra arguments.

Details

The distribution D is used to specify an initial distribution. The list prm contains details concerning a single parameter that is allowed to change values. The quantity of interest is evaluated as a function of this parameter.

Specifically, prm includes three elements named "name", "pos", and "val". The first two elements determine the exact parameter that changes, while the third one is a numeric vector holding the values it takes. For example, in the case of the Multivariate Gamma distribution, D <- MGamma(shape = c(1, 2), scale = 3) and prm <- list(name = "shape", pos = 2, val = seq(1, 1.5, by = 0.1)) means that the evaluation will be performed for the MGamma distributions with shape parameters ⁠(1, 1)⁠, ⁠(1, 1.1)⁠, ..., ⁠(1, 1.5)⁠ and scale 3. Notice that the initial shape parameter 2 in D is not utilized in the function.

Value

For the small sample, a data.frame with columns named "Parameter", "Observations", "Estimator", "Metric", and "Value". For the large sample, a data.frame with columns "Row", "Col", "Parameter", "Estimator", and "Value".

See Also

plot_small_metrics large_metrics, plot_large_metrics

Examples


D <- Beta(shape1 = 1, shape2 = 2)

prm <- list(name = "shape1",
            pos = NULL,
            val = seq(0.5, 2, by = 0.5))

x <- small_metrics(D, prm,
                   est = c("mle", "me", "same"),
                   obs = c(20, 50),
                   sam = 1e2,
                   seed = 1)

plot_small_metrics(x)