On the Distribution of Linear Combinations of Non-central Chi-Squares
David A. Harville · The Annals of Mathematical Statistics · 1971
Press [1] expressed the distribution of an arbitrary linear combination of noncentral chi-square variates as a mixture of distributions of weighted differences between pairs of central chi-squares. The distributions appearing in the mixture depend on the coefficients in the linear combination. Here, by modifying Press's results, we obtain a mixture representation not exhibiting that property. Following Press, let X4,d denote a non-central chi-square variate, having m degrees of freedom and non-centrality parameter d, whose probability density function is given by