3.13.5 Other Parameters of Quadratic Polynomials of Multivariate Normal Random Vectors

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:





where μ = E(Y). The variance of Y is, by our result from Exercise 3.15,


The skewness η1 and kurtosis η2 are obtained as



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.


 Consider random vector X ~ N3(μ,Σ) with








Express Y as a linear polynomial of independent chi-squared and normal random variables.


Calculate the mean and standard deviation of the random variable Y of the previous exercise.