GB2 (version 2.1.1)

Moments: 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.

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 \(log(X)\).

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.

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

# NOT RUN {
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)
# }