Moments: Moments and Other Properties of a GB2 Random Variable

In GB2: Generalized Beta Distribution of the Second Kind: Properties, Likelihood, Estimation

Moments and Other Properties of a GB2 Random Variable

Description

These functions calculate the moments of order k and incomplete moments of order k of a GB2 random variable X as well as the expectation, the variance, the kurtosis and the skewness of log(X).

Usage

moment.gb2(k, shape1, scale, shape2, shape3)
incompl.gb2(x, k, shape1, scale, shape2, shape3)
el.gb2(shape1, scale, shape2, shape3)
vl.gb2(shape1, shape2, shape3)
sl.gb2(shape2, shape3)
kl.gb2(shape2, shape3)

Arguments

x	numeric; vector of quantiles.
k	numeric; order of the moment.
shape1	numeric; positive parameter.
scale	numeric; positive parameter.
shape2, shape3	numeric; positive parameters of the Beta distribution.

Details

Let X be a random variable following a GB2 distribution with parameters shape1 = a, scale = b, shape2 = p and shape3 = q. Moments and incomplete moments of X exist only for -ap \le k \le aq. Moments are given by

E(X^k) = {b}^{k} \frac{\Gamma (p+k/a) \Gamma (q-k/a)}{\Gamma (p) \Gamma (q)}

This expression, when considered a function of k, can be viewed as the moment-generating function of Y=log(X). Thus, it is useful to compute the moments of log(X), which are needed for deriving, for instance, the Fisher information matrix of the GB2 distribution. Moments of log(X) exist for all k.

Value

moment.gb2 gives the moment of order k, incompl.gb2 gives the incomplete moment of order k, El.gb2 gives the expectation of log(X), vl.gb2 gives the variance of log(X), sl.gb2 gives the skewness of log(X), kl.gb2 gives the kurtosis of log(X).

Author(s)

Monique Graf

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, chapter 6. Wiley, Ney York.

See Also

gamma for the Gamma function and related functions (digamma, trigamma and psigamma).

Examples

a <- 3.9
b <- 18873
p <- 0.97
q <- 1.03
k <- 2
x <- qgb2(0.6, a, b, p, q)
moment.gb2(k, a, b, p, q)
incompl.gb2(x, k, a, b, p, q)
vl.gb2(a, p, q)
kl.gb2(p, q)

GB2 documentation built on Aug. 21, 2025, 6:03 p.m.