Type:	Package
Title:	Raw, Central and Standardized Moments of Parametric Distributions
Version:	0.1.3
Date:	2025-03-15
Description:	To calculate the raw, central and standardized moments from distribution parameters. To solve the distribution parameters based on user-provided mean, standard deviation, skewness and kurtosis. Normal, skew-normal, skew-t and Tukey g-&-h distributions are supported, for now.
License:	GPL-2
Encoding:	UTF-8
Language:	en-US
Depends:	R (≥ 4.4.0)
Imports:	methods
Suggests:	sn
RoxygenNote:	7.3.2
NeedsCompilation:	no
Packaged:	2025-03-15 18:17:07 UTC; tingtingzhan
Author:	Tingting Zhan [aut, cre]
Maintainer:	Tingting Zhan <tingtingzhan@gmail.com>
Repository:	CRAN
Date/Publication:	2025-03-15 22:10:05 UTC

Raw, Central and Standardized Moments, and other Distribution Characteristics

Description

Up to 4th raw \text{E}(Y^n), central \text{E}[(Y-\mu)^n] and standardized moments \text{E}[(Y-\mu)^n/\sigma^n] of the random variable

Y = (X - \text{location})/\text{scale}

Also, the mean, standard deviation, skewness and excess kurtosis of the random variable X.

Details

For Y = (X - \text{location})/\text{scale}, let \mu = \text{E}(Y), then, according to Binomial theorem, the 2nd to 4th central moments of Y are,

\text{E}[(Y-\mu)^2] = \text{E}(Y^2) - 2\mu \text{E}(Y) + \mu^2 = \text{E}(Y^2) - \mu^2

\text{E}[(Y-\mu)^3] = \text{E}(Y^3) - 3\mu \text{E}(Y^2) + 3\mu^2 \text{E}(Y) - \mu^3 = \text{E}(Y^3) - 3\mu \text{E}(Y^2) + 2\mu^3

\text{E}[(Y-\mu)^4] = \text{E}(Y^4) - 4\mu \text{E}(Y^3) + 6\mu^2 \text{E}(Y^2) - 4\mu^3 \text{E}(Y) + \mu^4 = \text{E}(Y^4) - 4\mu \text{E}(Y^3) + 6\mu^2 \text{E}(Y^2) - 3\mu^4

The distribution characteristics of Y are,

\mu_Y = \mu

\sigma_Y = \sqrt{\text{E}[(Y-\mu)^2]}

\text{skewness}_Y = \text{E}[(Y-\mu)^3] / \sigma^3_Y

\text{kurtosis}_Y = \text{E}[(Y-\mu)^4] / \sigma^4_Y - 3

The distribution characteristics of X are \mu_X = \text{location} + \text{scale}\cdot \mu_Y, \sigma_X = \text{scale}\cdot \sigma_Y, \text{skewness}_X = \text{skewness}_Y, and \text{kurtosis}_X = \text{kurtosis}_Y.

Slots

distname

character scalar, name of distribution, e.g., 'norm' for normal, 'sn' for skew-normal, 'st' for skew-t, and 'GH' for Tukey g-&-h distribution, following the nomenclature of dnorm, dsn, dst and QuantileGH::dGH

location,scale

numeric scalars or vectors, location and scale parameters

mu

numeric scalar or vector, 1st raw moment \mu = \text{E}(Y). Note that the 1st central moment \text{E}(Y-\mu) and standardized moment \text{E}(Y-\mu)/\sigma are both 0.

raw2,raw3,raw4

numeric scalars or vectors, 2nd or higher raw moments \text{E}(Y^n), n\geq 2

central2,central3,central4

numeric scalars or vectors, 2nd or higher central moments, \sigma^2 = \text{E}[(Y-\mu)^2] and \text{E}[(Y-\mu)^n], n\geq 3

standardized3,standardized4

numeric scalars or vectors, 3rd or higher standardized moments, skewness \text{E}[(Y-\mu)^3]/\sigma^3 and kurtosis \text{E}[(Y-\mu)^4]/\sigma^4. Note that the 2nd standardized moment is 1

Note

Potential name clash with function e1071::moment.

Solve Tukey g-&-h Parameters from Moments

Description

Solve Tukey g-, h- and g-&-h distribution parameters from mean, standard deviation, skewness and kurtosis.

Usage

moment2GH(mean = 0, sd = 1, skewness, kurtosis)

moment2GH_h_demo(sd = 1, kurtosis)

moment2GH_g_demo(mean = 0, sd = 1, skewness)

Arguments

Details

Function moment2GH() solves the location A, scale B, skewness g and elongation h parameters of Tukey g-&-h distribution, from user-specified mean \mu (default 0), standard deviation \sigma (default 1), skewness and kurtosis.

An educational and demonstration function moment2GH_h_demo() solves (B, h) parameters of Tukey h-distribution, from user-specified \sigma and kurtosis. This is a non-skewed distribution, thus the location parameter A=\mu=0, and the skewness parameter g=0.

An educational and demonstration function moment2GH_g_demo() solves (A, B, g) parameters of Tukey g-distribution, from user-specified \mu, \sigma and skewness. For this distribution, the elongation parameter h=0.

Value

Function moment2GH() returns a length-4 numeric vector (A, B, g, h).

Function moment2GH_h_demo() returns a length-2 numeric vector (B, h).

Function moment2GH_g_demo() returns a length-3 numeric vector (A, B, g).

Examples

moment2GH(skewness = .2, kurtosis = .3)

moment2GH_h_demo(kurtosis = .3)

moment2GH_g_demo(skewness = .2)

Moment to Parameters: A Batch Process

Description

Converts multiple sets of moments to multiple sets of distribution parameters.

Usage

moment2param(distname, FUN = paste0("moment2", distname), ...)

Arguments

Value

Function moment2param() returns a list of numeric vectors.

Examples

skw = c(.2, .5, .8)
krt = c(.5, 1, 1.5)
moment2param(distname = 'GH', skewness = skw, kurtosis = krt)
moment2param(distname = 'st', skewness = skw, kurtosis = krt)

Solve Skew-Normal Parameters from Moments

Description

Solve skew-normal parameters from mean, standard deviation and skewness.

Usage

moment2sn(mean = 0, sd = 1, skewness)

Arguments

Details

Function moment2sn() solves the location \xi, scale \omega and slant \alpha parameters of skew-normal distribution, from user-specified mean \mu (default 0), standard deviation \sigma (default 1) and skewness.

Value

Function moment2sn() returns a length-3 numeric vector (\xi, \omega, \alpha).

Examples

moment2sn(skewness = .3)

Solve Skew-t Parameters from Moments

Description

Solve skew-t parameters from mean, standard deviation, skewness and kurtosis.

Usage

moment2st(mean = 0, sd = 1, skewness, kurtosis)

moment2t_demo(sd = 1, kurtosis)

Arguments

Details

Function moment2st() solves the location \xi, scale \omega, slant \alpha and degree of freedom \nu parameters of skew-t distribution, from user-specified mean \mu (default 0), standard deviation \sigma (default 1), skewness and kurtosis.

An educational and demonstration function moment2t_demo solves (\omega, \nu) parameters of t-distribution, from user-specified \sigma and kurtosis. This is a non-skewed distribution, thus the location parameter \xi=\mu=0, and the slant parameter \alpha=0.

Value

Function moment2st() returns a length-4 numeric vector (\xi, \omega, \alpha, \nu).

Function moment2t_demo() returns a length-2 numeric vector (\omega, \nu).

Examples

moment2st(skewness = .2, kurtosis = .3)

moment2t_demo(kurtosis = .3)

Moments of Tukey g-&-h Distribution

Description

Moments of Tukey g-&-h distribution.

Usage

moment_GH(A = 0, B = 1, g = 0, h = 0)

Arguments

Value

Function moment_GH() returns a moment object.

References

Raw moments of Tukey g-&-h distribution: doi:10.1002/9781118150702.ch11

Examples

A = 3; B = 1.5; g = .7; h = .01
moment_GH(A = A, B = B, g = 0, h = h)
moment_GH(A = A, B = B, g = g, h = 0)
moment_GH(A = A, B = B, g = g, h = h)

Moments of Normal Distribution

Description

Moments of normal distribution, parameter nomenclature follows dnorm function.

Usage

moment_norm(mean = 0, sd = 1)

Arguments

Value

Function moment_norm() returns a moment object.

Examples

moment_norm(mean = 1.2, sd = .7)

Moments of Skew-Normal Distribution

Description

Moments of skew-normal distribution, parameter nomenclature follows dsn function.

Usage

moment_sn(xi = 0, omega = 1, alpha = 0)

Arguments

Value

Function moment_sn() returns a moment object.

Examples

xi = 2; omega = 1.3; alpha = 3
moment_sn(xi, omega, alpha)
curve(sn::dsn(x, xi = 2, omega = 1.3, alpha = 3), from = 0, to = 6)

Moments of Skew-t Distribution

Description

Moments of skew-t distribution, parameter nomenclature follows dst function.

Usage

moment_st(xi = 0, omega = 1, alpha = 0, nu = Inf)

Arguments

Value

Function moment_st() returns a moment object.

References

Raw moments of skew-t: https://arxiv.org/abs/0911.2342

Examples

xi = 2; omega = 1.3; alpha = 3; nu = 6
curve(sn::dst(x, xi = xi, omega = omega, alpha = alpha, nu = nu), from = 0, to = 6)
moment_st(xi, omega, alpha, nu)

Show moment

Description

Print S4 object moment in a pretty manner.

Usage

## S4 method for signature 'moment'
show(object)

Arguments

Value

The show method for moment object does not have a returned value.