Package 'estimators'

Asymptotic Variance

Description

These functions calculate 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, ...)

avar_mle(distr, ...)

avar_me(distr, ...)

avar_same(distr, ...)

Arguments

distr	A Distribution object.
type	character, case ignored. The estimator type ("mle", "me", or "same").
...	extra arguments.

Value

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

Functions

• avar_mle(): Asymptotic Variance of the Maximum Likelihood Estimator

• avar_me(): Asymptotic Variance of the Moment Estimator

• avar_same(): Asymptotic Variance of the Score-Adjusted Moment Estimator

References

General Textbooks

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

Beta and gamma distribution families

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

• 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
# -----------------------------------------------------

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Bern Distribution

Description

The Bernoulli distribution is a discrete probability distribution which takes the value 1 with probability $p$ and the value 0 with probability $1 - p$, where $0 \leq p \leq 1$.

Usage

Bern(prob = 0.5)

dbern(x, prob, log = FALSE)

pbern(x, prob, lower.tail = TRUE, log.p = FALSE)

qbern(x, prob, lower.tail = TRUE, log.p = FALSE)

rbern(n, prob)

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

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

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

## S4 method for signature 'Bern,numeric'
r(distr, n)

## 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)

llbern(x, prob)

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

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

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

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

vbern(prob, type = "mle")

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

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

Arguments

prob	numeric. The distribution parameter, within the (0, 1) interval.
log, log.p	logical. Should the logarithm of the probability be returned?
lower.tail	logical. If TRUE (default), probabilities are $P(X \leq x)$, otherwise $P(X > x)$.
n	numeric. The sample size.
distr, x	If both arguments coexist, distr is an object of class Bern and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Bern instead.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability mass function (PMF) of the Bernoulli distribution is given by:

$f(x; p) = p^x (1 - p)^{1 - x}, \quad p \in (0, 1), \quad x \in \{0, 1\}.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dbinom(), pbinom(), qbinom(), rbinom()

Examples

# -----------------------------------------------------
# Bernoulli Distribution Example
# -----------------------------------------------------

# Create the distribution
p <- 0.7
D <- Bern(p)
x <- c(0, 1)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbern(x, p)

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("bern", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbern(p, type = "mle")
vbern(p, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Beta Distribution

Description

The Beta distribution is an absolute continuous probability distribution with support $S = [0,1]$, parameterized by two shape parameters, $\alpha > 0$ and $\beta > 0$.

Usage

Beta(shape1 = 1, shape2 = 1)

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

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

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

## S4 method for signature 'Beta,numeric'
r(distr, n)

## 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 'Beta,numeric'
ll(distr, x)

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

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

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

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

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	numeric. The distribution parameters (positive real numbers).
distr, x	If both arguments coexist, distr is an object of class Beta and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Beta instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
par0, method, lower, upper	arguments passed to optim for the mle optimization.

Details

The probability density function (PDF) of the Beta distribution is given by:

$f(x; \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \quad \alpha\in\mathbb{R}_+, \, \beta\in\mathbb{R}_+,$

for $x \in S = [0, 1]$, where $B(\alpha, \beta)$ is the Beta function:

$B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

References

• 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.

• Papadatos, N. (2022), On point estimators for gamma and beta distributions, arXiv preprint arXiv:2205.10799.

See Also

Functions from the stats package: dbeta(), pbeta(), qbeta(), rbeta()

Examples

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

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Binom Distribution

Description

The binomial distribution is a discrete probability distribution which models the probability of having x successes in n independent Binomoulli trials with success probability p.

Usage

Binom(size = 1, prob = 0.5)

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

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

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

## S4 method for signature 'Binom,numeric'
r(distr, n)

## 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'
entro(x)

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

llbinom(x, size, prob)

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

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

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

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

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

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

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

Arguments

size, prob	numeric. The distribution parameters, size must be a positive integer and prob must be within the (0, 1) interval.
distr, x	If both arguments coexist, distr is an object of class Binom and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Binom instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability mass function (PMF) of the binomial distribution is given by:

$f(x; n, p) = \binom{n}{x} p^x (1 - p)^{n - x}, \quad N \in \mathbb{N}, \quad p \in (0, 1),$

with $x \in \{0, 1, \dots, N\}$.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dbinom(), pbinom(), qbinom(), rbinom()

Examples

# -----------------------------------------------------
# Binomial Distribution Example
# -----------------------------------------------------

# Create the distribution
N <- 10 ; p <- 0.7
D <- Binom(N, p)
x <- 0:N
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbinom(x, N, p)

ebinom(x, size = N, type = "mle")
ebinom(x, size = N, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

# ------------------
# As. Variance
# ------------------

vbinom(N, p, type = "mle")
vbinom(N, p, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Cat Distribution

Description

The Categorical distribution is a discrete probability distribution that describes the probability of a single trial resulting in one of $k$ possible categories. It is a generalization of the Bernoulli distribution and a special case of the multinomial distribution with $n = 1$.

Usage

Cat(prob = c(0.5, 0.5))

dcat(x, prob)

rcat(n, prob)

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

## S4 method for signature 'Cat,numeric'
r(distr, n)

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

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

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

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

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

llcat(x, prob)

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

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

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

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

vcat(prob, type = "mle")

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

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

Arguments

prob	numeric. The distribution parameter, a probability vector.
n	numeric. The sample size.
distr, x	If both arguments coexist, distr is an object of class Cat and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Cat instead.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
dim	numeric. The parameter dimension. See details.

Details

The probability mass function (PMF) of the categorical distribution is given by:

$f(x; p) = \prod_{i=1}^k p_i^{x_i},$

subject to $\sum_{i=1}^{k} x_i = n$.

The estimation of prob from a sample would by default return a vector of probabilities corresponding to the categories that appeared in the sample and 0 for the rest. However, the parameter dimension cannot be uncovered by the sample, it has to be provided separately. This can be done with the argument dim. If dim is not supplied, the dimension will be retrieved from the distr argument. Categories that did not appear in the sample will have 0 probabilities appended to the end of the prob vector.

Note that the actual dimension of the probability parameter vector is k-1, therefore the Fisher information matrix and the asymptotic variance - covariance matrix of the estimators is of dimension ⁠(k-1)x(k-1)⁠.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

dmultinom(), rmultinom()

Examples

# -----------------------------------------------------
# Categorical Distribution Example
# -----------------------------------------------------

# Create the distribution
p <- c(0.1, 0.2, 0.7)
D <- Cat(p)
x <- 2
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
mode(D) # Mode
var(D) # Variance
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llcat(x, p)

ecat(x, dim = 3, type = "mle")
ecat(x, dim = 3, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("cat", dim = 3, x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vcat(p, type = "mle")
vcat(p, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Cauchy Distribution

Description

The Cauchy distribution is an absolute continuous probability distribution characterized by its location parameter $x_0$ and scale parameter $\gamma > 0$.

Usage

Cauchy(location = 0, scale = 1)

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

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

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

## S4 method for signature 'Cauchy,numeric'
r(distr, n)

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

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

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

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

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

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

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

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

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

llcauchy(x, location, scale)

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

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

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

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

vcauchy(location, scale, type = "mle")

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

Arguments

location, scale	numeric. The distribution parameters.
distr, x	If both arguments coexist, distr is an object of class Cauchy and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Cauchy instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
par0, method, lower, upper	arguments passed to optim for the mle optimization.

Details

The probability density function (PDF) of the Cauchy distribution is given by:

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

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dcauchy(), pcauchy(), qcauchy(), rcauchy()

Examples

# -----------------------------------------------------
# Cauchy Distribution Example
# -----------------------------------------------------

# Create the distribution
x0 <- 3 ; scale <- 5
D <- Cauchy(x0, scale)
x <- c(-5, 3, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

median(D) # Median
mode(D) # Mode
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llcauchy(x, x0, scale)

ecauchy(x, type = "mle")
ecauchy(x, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("cauchy", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vcauchy(x0, scale, type = "mle")
avar_mle(D)
avar(D, type = "mle")

---

Chi-Square Distribution

Description

The Chi-Square distribution is a continuous probability distribution commonly used in statistical inference, particularly in hypothesis testing and confidence interval estimation. It is defined by the degrees of freedom parameter $k > 0$.

Usage

Chisq(df = 1)

## S4 method for signature 'Chisq,numeric'
d(distr, x)

## S4 method for signature 'Chisq,numeric'
p(distr, x)

## S4 method for signature 'Chisq,numeric'
qn(distr, x)

## S4 method for signature 'Chisq,numeric'
r(distr, n)

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

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

## S4 method for signature 'Chisq'
mode(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)

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

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

llchisq(x, df)

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

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

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

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

vchisq(df, type = "mle")

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

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

Arguments

df	numeric. The distribution degrees of freedom parameter.
distr, x	If both arguments coexist, distr is an object of class Chisq and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Chisq instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability density function (PDF) of the Chi-Square distribution is given by:

$f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dchisq(), pchisq(), qchisq(), rchisq()

Examples

# -----------------------------------------------------
# Chi-Square Distribution Example
# -----------------------------------------------------

# Create the distribution
df <- 4
D <- Chisq(df)
x <- c(0.3, 2, 20)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
den <- d(D) ; den(x) # den is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llchisq(x, df)

echisq(x, type = "mle")
echisq(x, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("chisq", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vchisq(df, type = "mle")
vchisq(df, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Dirichlet Distribution

Description

The Dirichlet distribution is an absolute continuous probability, specifically a multivariate generalization of the beta distribution, parameterized by a vector $\boldsymbol{\alpha} = (\alpha_1, \alpha_2, ..., \alpha_k)$ with $\alpha_i > 0$.

Usage

Dir(alpha = c(1, 1))

ddir(x, alpha, log = FALSE)

rdir(n, alpha)

## S4 method for signature 'Dir,numeric'
d(distr, x)

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

## S4 method for signature 'Dir,numeric'
r(distr, n)

## 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)

lldirichlet(x, alpha)

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

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

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

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

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

vdirichlet(alpha, type = "mle")

## 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 parameter vector.
log	logical. Should the logarithm of the density be returned?
n	numeric. The sample size.
distr, x	If both arguments coexist, distr is an object of class Dir and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Dir instead.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
par0, method, lower, upper	arguments passed to optim for the mle optimization.

Details

The probability density function (PDF) of the Dirichlet distribution is given by:

$f(x_1, ..., x_k; \alpha_1, ..., \alpha_k) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^k x_i^{\alpha_i - 1},$

where $B(\boldsymbol{\alpha})$ is the multivariate Beta function:

$B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)}$

and $\sum_{i=1}^k x_i = 1$, $x_i > 0$.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

References

• Oikonomidis, I. & Trevezas, S. (2025), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

Examples

# -----------------------------------------------------
# Dir Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- c(0.5, 2, 5)
D <- Dir(a)
x <- c(0.3, 0.2, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
mode(D) # Mode
var(D) # Variance
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
lldirichlet(x, a)

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("dir", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vdirichlet(a, type = "mle")
vdirichlet(a, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Distribution Functionals

Description

A collection of S4 classes that provide a flexible and structured way to work with probability distributions.

Usage

## S4 method for signature 'Distribution,missing'
d(distr, x, ...)

## S4 method for signature 'Distribution,missing'
p(distr, x, ...)

## S4 method for signature 'Distribution,missing'
qn(distr, x, ...)

## S4 method for signature 'Distribution,missing'
r(distr, n, ...)

## S4 method for signature 'Distribution,missing'
ll(distr, x, ...)

## S4 method for signature 'Distribution,missing'
mle(distr, x, ...)

## S4 method for signature 'Distribution,missing'
me(distr, x, ...)

## S4 method for signature 'Distribution,missing'
same(distr, x, ...)

Arguments

distr	a Distribution object.
x, n	missing. Arguments not supplied.
...	extra arguments.

Details

When x or n are missing, the methods return a function that takes as input the missing argument, allowing the user to work with the function object itself. See examples.

Value

When supplied with one argument, the d(), p(), q(), r() ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively.

See Also

moments, loglikelihood, estimation, Bern, Beta, Binom, Cat, Cauchy, Chisq, Dir, Exp, Fisher, Gam, Geom, Laplace, Lnorm, Multigam, Multinom, Nbinom, Norm, Pois, Stud, Unif, Weib

Examples

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

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Distribution S4 Classes

Description

A collection of S4 classes that provide a flexible and structured way to work with probability distributions.

Usage

d(distr, x, ...)

p(distr, x, ...)

qn(distr, x, ...)

r(distr, n, ...)

Arguments

distr	a Distribution object.
x	numeric. The point to evaluate the function.
...	extra arguments.
n	numeric. The sample size.

Details

These S4 generic methods can work both as functions and as functionals (functions that return functions). The available distribution families are coded as S4 classes, specifically subclasses of the Distribution superclass. The methods can be used in two ways:

Option 1: If both the distr argument and x or n are supplied, then the function is evaluated directly, as usual.

Option 2: If only the distr argument is supplied, the method returns a function that takes as input the missing argument x or n, allowing the user to work with the function object itself. See examples.

Looking for a specific distribution family? This help page is general. Use the help page of each distribution to see the available methods for the class, details, and examples. Check the See Also section.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Functions

• d(): density function

• p(): cumulative distribution function

• qn(): generalized inverse distribution function

• r(): random sample generator function

See Also

moments, loglikelihood, estimation, Bern, Beta, Binom, Cat, Cauchy, Chisq, Dir, Exp, Fisher, Gam, Geom, Laplace, Lnorm, Multigam, Multinom, Nbinom, Norm, Pois, Stud, Unif, Weib

Examples

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

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Parameter Estimation

Description

This set of functions estimates the parameters of a random sample according to a specified family of distributions. See details.

Usage

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

mle(distr, x, ...)

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

me(distr, x, ...)

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

same(distr, x, ...)

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

Arguments

distr	A Distribution object or a character. The distribution family assumed.
x	numeric. A sample under estimation.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The package covers three major estimation methods: maximum likelihood estimation (MLE), moment estimation (ME), and score-adjusted estimation (SAME).

In order to perform parameter estimation, a new ⁠e<name>()⁠ member is added to the d(), p(), q(), r() family, following the standard stats name convention. These functions take two arguments, the observations x (an atomic vector for univariate or a matrix for multivariate distibutions) and the type of estimation method to use (a character with possible values "mle", "me", and "same".)

Point estimation functions are available in two versions, the distribution specific one, e.g. ebeta(), and the S4 generic ones, namely mle(), me(), and same(). A general function called e() is also implemented, covering all distributions and estimators.

Value

list. The estimator of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

Functions

• mle(): Maximum Likelihood Estimator

• me(): Moment Estimator

• same(): Score - Adjusted Moment Estimation

References

General Textbooks

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

Beta and gamma distribution families

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

• 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
# -----------------------------------------------------

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Exponential Distribution

Description

The Exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant average rate. It is defined by the rate parameter $\lambda > 0$.

Usage

Exp(rate = 1)

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

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

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

## S4 method for signature 'Exp,numeric'
r(distr, n)

## 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)

llexp(x, rate)

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

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

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

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

vexp(rate, type = "mle")

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

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

Arguments

rate	numeric. The distribution rate parameter.
distr, x	If both arguments coexist, distr is an object of class Exp and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Exp instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability density function (PDF) of the Exponential distribution is given by:

$f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0 .$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dexp(), pexp(), qexp(), rexp()

Examples

# -----------------------------------------------------
# Exp Distribution Example
# -----------------------------------------------------

# Create the distribution
rate <- 5
D <- Exp(rate)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llexp(x, rate)

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("exp", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vexp(rate, type = "mle")
vexp(rate, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Fisher Distribution

Description

The Fisher (F) distribution is an absolute continuous probability distribution that arises frequently in the analysis of variance (ANOVA) and in hypothesis testing. It is defined by two degrees of freedom parameters $d_1 > 0$ and $d_2 > 0$.

Usage

Fisher(df1 = 1, df2 = 1)

## S4 method for signature 'Fisher,numeric'
d(distr, x)

## S4 method for signature 'Fisher,numeric'
p(distr, x)

## S4 method for signature 'Fisher,numeric'
qn(distr, x)

## S4 method for signature 'Fisher,numeric'
r(distr, n)

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

## S4 method for signature 'Fisher'
median(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)

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

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

llf(x, df1, df2)

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

Arguments

df1, df2	numeric. The distribution parameters.
distr, x	If both arguments coexist, distr is an object of class Fisher and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Fisher instead.
n	numeric. The sample size.

Details

The probability density function (PDF) of the F-distribution is given by:

$f(x; d_1, d_2) = \frac{\sqrt{\left(\frac{d_1 x}{d_1 x + d_2}\right)^{d_1} \left(\frac{d_2}{d_1 x + d_2}\right)^{d_2}}}{x B(d_1/2, d_2/2)}, \quad x > 0 .$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: df(), pf(), qf(), rf()

Examples

# -----------------------------------------------------
# Fisher Distribution Example
# -----------------------------------------------------

# Create the distribution
df1 <- 14 ; df2 <- 20
D <- Fisher(df1, df2)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llf(x, df1, df2)

---

Gamma Distribution

Description

The Gamma distribution is an absolute continuous probability distribution with two parameters: shape $\alpha > 0$ and scale $\beta > 0$.

Usage

Gam(shape = 1, scale = 1)

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

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

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

## S4 method for signature 'Gam,numeric'
r(distr, n)

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

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

## S4 method for signature 'Gam'
mode(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)

llgamma(x, shape, scale)

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

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

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

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

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

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

## 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 (positive real numbers).
distr, x	If both arguments coexist, distr is an object of class Gamma and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Gamma instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
par0, method, lower, upper	arguments passed to optim for the mle optimization.

Details

The probability density function (PDF) of the Gamma distribution is given by:

$f(x; \alpha, \beta) = \frac{\beta^{-\alpha} x^{\alpha-1} e^{-x/\beta}}{\Gamma(\alpha)}, \quad x > 0.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

References

• 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.

• Papadatos, N. (2022), On point estimators for gamma and beta distributions, arXiv preprint arXiv:2205.10799.

See Also

Functions from the stats package: dgamma(), pgamma(), qgamma(), rgamma()

Examples

# -----------------------------------------------------
# Gamma Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- 3 ; b <- 5
D <- Gam(a, b)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llgamma(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("gam", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vgamma(a, b, type = "mle")
vgamma(a, b, type = "me")
vgamma(a, b, type = "same")

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

avar(D, type = "mle")

---

Geometric Distribution

Description

The Geometric distribution is a discrete probability distribution that models the number of failures before the first success in a sequence of independent Bernoulli trials, each with success probability $0 < p \leq 1$.

Usage

Geom(prob = 0.5)

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

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

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

## S4 method for signature 'Geom,numeric'
r(distr, n)

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

## S4 method for signature 'Geom'
median(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)

llgeom(x, prob)

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

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

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

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

vgeom(prob, type = "mle")

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

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

Arguments

prob	numeric. The distribution parameter.
distr, x	If both arguments coexist, distr is an object of class Geom and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Geom instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability mass function (PMF) of the Geometric distribution is:

$P(X = k) = (1 - p)^k p, \quad k \in \mathbb{N}_0.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dgeom(), pgeom(), qgeom(), rgeom()

Examples

# -----------------------------------------------------
# Geom Distribution Example
# -----------------------------------------------------

# Create the distribution
p <- 0.4
D <- Geom(p)
x <- 0:4
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llgeom(x, p)

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("geom", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vgeom(p, type = "mle")
vgeom(p, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Polygamma Functions

Description

This set of functions revolve around the polygamma functions, i.e. the derivatives of the gamma function.

Usage

idigamma(x, ...)

Ddigamma(x, y)

Dtrigamma(x, y)

gammap(x, p, log = FALSE)

Arguments

x, y	numeric. The points to evaluate the function.
...	extra arguments passed to optim().
p	integer. The p-variate Gamma function.
log	logical. Should the logarithm of the result be returned?

Details

These functions are needed for the beta and gamma distribution families (and their multivariate analogs, e.g. the Dirichlet). They appear in the estimation and the asymptotic variance-covariance matrix of the MLE and the SAME.

The idigamma() function implements the inverse of the digamma function $\psi$. It is a numerical approximation based on the L-BFGS-U quasi-Newton algorithm. Specifically, idigamma() makes a call to optim() in order to to solve the equation $\psi(x) = y$; more accurately, to find the minimum of $f(x) = \log\Gamma(x) - xy$, whose derivative is $f'(x) = \psi(x) - y$. The optimization is restricted within the tight bounds derived by Batir (2017). The function is vectorized.

Value

numeric. The evaluated function.

Functions

• idigamma(): inverse digamma function.

• Ddigamma(): digamma difference function.

• Dtrigamma(): trigamma difference function.

• gammap(): p-variate gamma function

References

Necdet Batir (2017), INEQUALITIES FOR THE INVERSES OF THE POLYGAMMA FUNCTIONS https://arxiv.org/pdf/1705.06547

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

Examples

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

---

Laplace Distribution

Description

The Laplace distribution, also known as the double exponential distribution, is a continuous probability distribution that is often used to model data with sharp peaks and heavy tails. It is parameterized by a location parameter $\mu$ and a scale parameter $b > 0$.

Usage

Laplace(mu = 0, sigma = 1)

dlaplace(x, mu, sigma)

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

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

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

## S4 method for signature 'Laplace,numeric'
r(distr, n)

## 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)

lllaplace(x, mu, sigma)

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

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

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

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

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

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

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

Arguments

mu, sigma	numeric. The distribution parameters.
distr, x	If both arguments coexist, distr is an object of class Laplace and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Laplace instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability density function (PDF) of the Laplace distribution is:

$f(x; \mu, b) = \frac{1}{2b} \exp\left(-\frac{|x - \mu|}{b}\right) .$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Examples

# -----------------------------------------------------
# Laplace Distribution Example
# -----------------------------------------------------

# Create the distribution
m <- 3 ; s <- 5
D <- Laplace(m, s)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("laplace", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vlaplace(m, s, type = "mle")
vlaplace(m, s, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Large Sample Metrics

Description

This function calculates the asymptotic variance - covariance matrix characterizing the large sample (asymptotic) behavior of an estimator. The function evaluates the metrics as a function of a single parameter, keeping the other ones constant. See Details.

Usage

LargeMetrics(D, est, df)

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

Arguments

D	A subclass of Distribution. The distribution family of interest.
est	character. The estimator of interest. Can be a vector.
df	data.frame. a data.frame with columns named "Row", "Col", "Parameter", "Estimator", and "Value".
prm	A list containing three elements (name, pos, val). See Details.
...	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.

The prm list includes two elements named "name" and "val". The first one specifies the parameter that changes, and the second one is a numeric vector holding the values it takes.

In case the parameter of interest is a vector, a third element named "pos" can be specified to indicate the exact paramater that changes. In the example shown below, the evaluation will be performed for the Dirichlet distributions with shape parameters ⁠(0.5, 1)⁠, ⁠(0.6, 1)⁠, ..., ⁠(2, 1)⁠. Notice that the initial shape parameter value (1) is not utilized in the function.

Value

An object of class LargeMetrics with slots D, est, and df.

See Also

SmallMetrics, PlotMetrics

Examples

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

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

prm <- list(name = "shape1",
            val = seq(0.5, 2, by = 0.1))

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

plot(x)

# -----------------------------------------------------
# Dirichlet Distribution Example
# -----------------------------------------------------

D <- Dir(alpha = 1:2)

prm <- list(name = "alpha",
            pos = 1,
            val = seq(0.5, 2, by = 0.1))

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

plot(x)

---

Log-Normal Distribution

Description

The Lognormal distribution is an absolute continuous probability distribution of a random variable whose logarithm is normally distributed. It is defined by parameters $\mu$ and $\sigma > 0$, which are the mean and standard deviation of the underlying normal distribution.

Usage

Lnorm(meanlog = 0, sdlog = 1)

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

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

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

## S4 method for signature 'Lnorm,numeric'
r(distr, n)

## 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)

lllnorm(x, meanlog, sdlog)

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

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

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

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

vlnorm(meanlog, sdlog, type = "mle")

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

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

Arguments

meanlog, sdlog	numeric. The distribution parameters.
distr, x	If both arguments coexist, distr is an object of class Lnorm and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Lnorm instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability density function (PDF) of the Lognormal distribution is:

$f(x; \mu, \sigma) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\log x - \mu)^2}{2 \sigma^2}}, \quad x > 0 .$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dlnorm(), plnorm(), qlnorm(), rlnorm()

Examples

# -----------------------------------------------------
# Lnorm Distribution Example
# -----------------------------------------------------

# Create the distribution
m <- 3 ; s <- 5
D <- Lnorm(m, s)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("lnorm", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vlnorm(m, s, type = "mle")
vlnorm(m, s, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---

Log-Likelihood Function

Description

This function calculates the log-likelihood of an independent and identically distributed (iid) sample from a distribution. See Details.

Usage

ll(distr, x, ...)

Arguments

distr	A Distribution object
x	numeric. A sample under estimation.
...	extra arguments.

Details

The log-likelihood functions are provided in two forms: the ⁠ll<name>⁠ distribution-specific version that follows the stats package conventions, and the S4 generic ll. Examples for the ⁠ll<name>⁠ version are included in the distribution-specific help pages, e.g. ?Beta (all distributions can be found in the See Also section of the Distributions help page).

As with the d(), p(), q(), r() methods, ll() can be supplied only with distr to return the log-likelihood function (i.e. it can be used as a functional), or with both distr and x to be evaluated directly.

In some distribution families like beta and gamma, the MLE cannot be explicitly derived and numerical optimization algorithms have to be employed. Even in “good" scenarios, with plenty of observations and a smooth optimization function, extra care should be taken to ensure a fast and right convergence if possible. Two important steps are taken in package in this direction:

1. The log-likelihood function is analytically calculated for each distribution family, so that constant terms with respect to the parameters can be removed, leaving only the sufficient statistics as a requirement for the function evaluation.

2. Multidimensional problems are reduced to unidimensional ones by utilizing the score equations.

The resulting function that is inserted in the optimization algorithm is called lloptim(), and is not to be confused with the actual log-likelihood function ll(). The corresponding derivative is called dlloptim(). These functions are used internally and are not exported.

Therefore, whenever numerical computation of the MLE is required, optim() is called to optimize lloptim(), using the ME or SAME as the starting point (user's choice), and the L-BFGS-U optimization algorithm, with lower and upper limits defined by default as the parameter space boundary. Illustrative examples can be found in the package vignette.

Value

If only the distr argument is supplied, ll() returns a function. If both distr and x are supplied, ll() returns a numeric, the value of the log-likelihood function.

See Also

Distributions, moments, estimation

Examples

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

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, 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	a Distribution object.
...	extra arguments.
y, use, na.rm	arguments in mean and var standard methods from the stats package not used here.

Details

Given a distribution, these functions calculate the theoretical moments and other parametric quantities of interest. Some distribution-function combinations are not available; for example, the sd() function is available only for univariate distributions.

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

Technical Note: The argument of the moment functions does not follow the naming convention of the package, i.e. the Distribution object is names x rather than distr. This is due to the fact that most of the generics are already defined in the stats package (mean, median, mode, var, sd), therefore the first argument was already named x and could not change.

Value

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

Functions

• median(): Median

• mode(): Mode

• var(): Variance

• sd(): Standard Deviation

• skew(): Skewness

• kurt(): Kurtosis

• entro(): Entropy

• finf(): Fisher Information (numeric or matrix)

See Also

Distributions, loglikelihood, estimation

Examples

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

# Create the distribution
a <- 3 ; b <- 5
D <- Beta(a, b)
x <- c(0.3, 0.8, 0.5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
p(D, x) # distribution function
qn(D, 0.8) # inverse distribution function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llbeta(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("beta", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vbeta(a, b, type = "mle")
vbeta(a, b, type = "me")
vbeta(a, b, type = "same")

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

avar(D, type = "mle")

---

Multivariate Gamma Distribution

Description

The multivariate gamma distribution is a multivariate absolute continuous probability distribution, defined as the cumulative sum of independent gamma random variables with possibly different shape parameters \alpha_i > 0, i\in\[k \] and the same scale $\beta > 0$.

Usage

Multigam(shape = 1, scale = 1)

dmultigam(x, shape, scale, log = FALSE)

rmultigam(n, shape, scale)

## S4 method for signature 'Multigam,numeric'
d(distr, x)

## S4 method for signature 'Multigam,matrix'
d(distr, x)

## S4 method for signature 'Multigam,numeric'
r(distr, n)

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

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

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

llmultigam(x, shape, scale)

## S4 method for signature 'Multigam,matrix'
ll(distr, x)

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

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

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

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

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

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

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

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

Arguments

shape, scale	numeric. The distribution parameters.
log	logical. Should the logarithm of the density be returned?
n	numeric. The sample size.
distr, x	If both arguments coexist, distr is an object of class Gamma and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Gamma instead.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.
par0, method, lower, upper	arguments passed to optim for the mle optimization.

Details

The probability density function (PDF) of the multivariate gamma distribution is given by:

$f(x; \alpha, \beta) = \frac{\beta^{-\alpha_0}}{\prod_{i=1}^k\Gamma(\alpha_i)}, e^{-x_k/\beta} x_1^{\alpha_1-1}\prod_{i=1}^k (x_i - x_{i-1})^{(\alpha_i-1)} \quad x > 0.$

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

References

• Oikonomidis, I. & Trevezas, S. (2025), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

Examples

# -----------------------------------------------------
# Multivariate Gamma Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- c(0.5, 3, 5) ; b <- 5
D <- Multigam(a, b)
x <- c(0.3, 2, 10)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function
x <- r(D, n) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llmultigam(x, a, b)

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

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("multigam", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vmultigam(a, b, type = "mle")
vmultigam(a, b, type = "me")
vmultigam(a, b, type = "same")

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

avar(D, type = "mle")

---

Multinomial Distribution

Description

The multinomial distribution is a discrete probability distribution which models the probability of having x successes in n independent categorical trials with success probability vector p.

Usage

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

## S4 method for signature 'Multinom,numeric'
d(distr, x)

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

## S4 method for signature 'Multinom,numeric'
r(distr, n)

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

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

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

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

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

llmultinom(x, size, prob)

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

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

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

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

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

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

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

Arguments

size, prob	numeric. The distribution parameters, size must be a positive integer and prob must be a probability vector.
distr, x	If both arguments coexist, distr is an object of class Multinom and x is a numeric vector, the sample of observations. For the moment functions that only take an x argument, x is an object of class Multinom instead.
n	numeric. The sample size.
type	character, case ignored. The estimator type (mle, me, or same).
...	extra arguments.

Details

The probability mass function (PMF) of the Multinomial distribution is:

$P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} \prod_{i=1}^k p_i^{x_i},$

subject to $\sum_{i=1}^{k} x_i = n$.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dmultinom(), rmultinom()

Examples

# -----------------------------------------------------
# Multinomial Distribution Example
# -----------------------------------------------------

# Create the distribution
N <- 10 ; p <- c(0.1, 0.2, 0.7)
D <- Multinom(N, p)
x <- c(2, 3, 5)
n <- 100

# ------------------
# dpqr Functions
# ------------------

d(D, x) # density function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

x <- r(D, n) # random generator function

# ------------------
# Moments
# ------------------

mean(D) # Expectation
mode(D) # Mode
var(D) # Variance
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llmultinom(x, N, p)

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

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("multinom", x) # the distr argument can be a character

# ------------------
# As. Variance
# ------------------

vmultinom(N, p, type = "mle")
vmultinom(N, p, type = "me")

avar_mle(D)
avar_me(D)

avar(D, type = "mle")

---