Moments {GB2} | R Documentation |
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)
.
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)
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. |
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
.
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)
.
Monique Graf
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, chapter 6. Wiley, Ney York.
gamma
for the Gamma function and related functions (digamma
, trigamma
and psigamma
).
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)