3.13.5 Other Parameters
We can calculate any central moment of Y. This is simply a matter of multiplying out the formula for the desired central moment and substituting in values for moments. Consider the third central moment:
[3.188]
[3.189]
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[3.191]
where μ = E(Y). The variance of Y is, by our result from Exercise 3.15,
[3.192]
The skewness η1 and kurtosis η2 are obtained as
[3.193]
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Quantiles of Y can be approximated using the Cornish-Fisher (1937) expansion, which we discuss in the next section. They can be calculated exactly using the inversion theorem that we discuss in Section 3.16.
Exercises
Consider random vector X ~ N3(μ,Σ) with
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Let
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where
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Express Y as a linear polynomial of independent chi-squared and normal random variables.
Solution
Calculate the mean and standard deviation of the random variable Y of the previous exercise.
Solution