Package 'param2moment'

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

A	numeric scalar or vector, location parameter $A$
B	numeric scalar or vector, scale parameter $B$
g	numeric scalar or vector, skewness parameter $g$
h	numeric scalar or vector, elongation parameter $h$

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)

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Moments of Normal Distribution

Description

Moments of normal distribution, parameter nomenclature follows dnorm function.

Usage

moment_norm(mean = 0, sd = 1)

Arguments

mean	numeric scalar or vector, mean parameter $\mu$
sd	numeric scalar or vector, standard deviation $\sigma$

Value

Function moment_norm returns a moment object.

Examples

moment_norm(mean = 1.2, sd = .7)

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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

xi	numeric scalar or vector, location parameter $\xi$
omega	numeric scalar or vector, scale parameter $\omega$
alpha	numeric scalar or vector, slant parameter $\alpha$

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)

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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

xi	numeric scalar or vector, location parameter $\xi$
omega	numeric scalar or vector, scale parameter $\omega$
alpha	numeric scalar or vector, slant parameter $\alpha$
nu	numeric scalar or vector, degree of freedom $\nu$

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)

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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.

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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

mean	numeric scalar, mean $\mu$, default value 0
sd	numeric scalar, standard deviation $\sigma$, default value 1
skewness	numeric scalar
kurtosis	numeric scalar

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)

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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

distname	character scalar, distribution name. Currently supported are 'GH' for Tukey $g$-&-$h$ distribution, 'sn' for skew-normal distribution and 'st' for skew-$t$ distribution
FUN	name or character scalar, (name of) function used to solve the distribution parameters from moments. Default is paste0('moment2', distname), e.g., moment2GH will be used for distname = 'GH'. To use one of the educational functions, specify FUN = moment2GH_g_demo or FUN = 'moment2GH_g_demo'.
...	numeric scalars, some or all of mean, sd, skewness and kurtosis (length will be recycled).

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)

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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

mean	numeric scalar, mean $\mu$, default value 0
sd	numeric scalar, standard deviation $\sigma$, default value 1
skewness	numeric scalar

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)

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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

mean	numeric scalar, mean $\mu$, default value 0
sd	numeric scalar, standard deviation $\sigma$, default value 1
skewness	numeric scalar
kurtosis	numeric scalar

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)

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Show moment

Description

Print S4 object moment in a pretty manner.

Usage

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

Arguments

object

a moment object

Value

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

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