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Kashif Ali
Kashif Ali

Reputation: 197

Inverse Moments of a Non-Central Chi-Square Distribution

I want to compute inverse moments and truncated inverse moments of a non-central chi-square distribution in R. How can I do that in R?

Suppose X follows the non-central chi-square distribution with degrees of freedom "k" and non-centrality parameter "t". My problem is to numerically compute the following expectations for various values of "t" so I can simulate the risk of James-Stein type estimators.

(i)   E[X^(-1)] and E[X^(-2)]
(ii)  E[X^(-1)I(A)] where I(A) is an indicator function of set A
(iii) E[1-c{X^(-2)}I(A)] where c is a constant.

Upvotes: 1

Views: 911

Answers (2)

steveo'america
steveo'america

Reputation: 206

Paolella's book, Intermediate Probability gives the moments of the non-central chi-square to various powers. See equation (10.10). You can find R code for these in the sadists package.

Upvotes: 0

josliber
josliber

Reputation: 44340

In general, you can numerically compute the expected value of a random variable by drawing a large number of samples and then averaging them. For instance, you could estimate the expected values of X^(-1) and X^(-2) with something like:

mean(rchisq(1000000, df=3, ncp=10)^-1)
# [1] 0.1152163
mean(rchisq(1000000, df=3, ncp=10)^-2)
# [1] 0.1371877

Upvotes: 2

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