r_moment_gamma_Weib {RSizeBiased}

r-th moment of the gamma or the Weibull distribution.

Description

Calculates the r-th moment of the gamma or Weibull distribution.

Usage

r_moment_gamma_Weib(TRpar,r,dist)

Arguments

TRpar

A vector of length 2, containing the shape and scale parameters of the distribution.

r

The size (order) of the distribution. The special cases r=1,2,3 correspond to length, area, volume biased samples respectively and are the most frequently encountered in practice. The case r=0 corresponds to random samples from the Gamma distribution.

dist

Character switch, enables the choice of distribution: type "weib" for the Weibull or "gamma" for the gamma distribution.

Details

In the case of the \Gamma(\alpha, \beta) distribution the r-th moment is given by

\mu_r = \int_0^{\infty} x^r f(x;\alpha, \beta)\,dx =\beta^r \frac{\Gamma(\alpha+r)}{\Gamma(\alpha)}, \alpha> -r

while for the W(\alpha, \beta) distribution the r-th moment is given by

\mu_r = \int_0^{\infty} x^r f(x;\alpha, \beta)\,dx = \beta^r \Gamma\left(1+\frac{\alpha}{r}\right), \alpha> -r

Value

A scalar, the value of the moment.

Author(s)

Polychronis Economou

R implementation and documentation: Polychronis Economou <[e-mail]>

References

Economou et. al. (2021). Hypothesis testing for the population mean and variance based on r-size biased samples, under review.

Examples

#r-moment for the Gamma distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "gamma")
#r-moment for for the Weibull distribution for true parms=(2,3), r=1:
r_moment_gamma_Weib(c(2,3),1, "weib")