moment-class {param2moment}

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.