Series Representations of Distributions of Quadratic Forms in Normal Variables II. Non-Central Case

Abstract

Abstract : The probability density function (pdf) of a positive definite quadratic form in (central or non-central) normal variables can be represented as a series expansion in a number of different ways. Among these, one of the most important is that of a series of pdf's of non-central chi-square's or central chi-square's with increasing degrees of freedom. These expansions have been discussed by Ruben (Ann. Math. Statist. 33 (1962) 542-570) (Ann. Math. Statist. 34 (1963) 1582-1584) who has given convenient recurrence formulae for determining the coefficients. Expansion in terms of Laguerre series and Maclaurin series (powers of the argument) have been discussed for central variables by Gurland (Ann. Math. Statist. 24 (1953) 416-427) and Pachares (Ann. Math. Statist. 26 (1955) 128-131) respectively, and in the general (non-central) case by Shah (Ann. Math. Statist. 34 (1963) 186-190) and Shah and Khatri (Ann. Math. Statist. 32 (1961) 883-887), but the coefficients in their series are not presented in a very convenient form for calculations. It is the purpose of this paper to show how all three kinds of expansion can be derived in a similar way, and incidentally, to obtain convenient recurrence formulae for determining the coefficients in the Laguerre and Maclaurin expansions. In the present paper the central case is discussed. (Author)