moment-class: 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 $X$.

Arguments

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

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 $\text{kurtosis}_X = \text{kurtosis}_Y$.