A New Result on the Distribution of Quadratic Forms

Abstract

The distribution of the non-homogeneous quadratic form $Q = \sum^n_1 a_i(x_i - b_i)^2$, where the $x_i$ are independent standardized normal variables and the $a_i$ and $b_i$ are real constants with $a_i > 0$, has recently [1] been obtained as an infinite linear combination in scaled central and noncentral $\chi^2$ distribution functions in the form \begin{equation*}\tag{1.1}P\lbrack Q \leqq t\rbrack = \sum^\infty_0 c_j(p) F_{n+2j}(t/p) = \sum^\infty_0 d_j(p) G_{n+2j;k}(t/p).\end{equation*} Here $p$ is an arbitrary positive constant, $F_{n + 2j}(\cdot)$ is the distribution function of $\chi^2$ with $n + 2j$ degrees of freedom and $G_{n + 2j;k}(\cdot)$ is the distribution function of $\chi^2$ with $n + 2j$ degrees of freedom and non-centrality parameter $\kappa = (\sum^n_1 b^2_i)^{\frac{1}{2}}.$ The main purpose of the present paper is to rederive the first of the two expansions in (1.1), for the special case $p \leqq \min_ia_i$ when the expansion is a proper mixture representation, by a simple conditional probability argument which may be of some general interest. At the same time the $c_j(p)$ will be expressed in simpler and more appealing form than in [1]. In essence, the distribution of $Q$ (including that of $\sum^n_1 a_ix^2_i$) is found to be almost a direct consequence of the distribution of the special non-homogeneous form $\sum^n_1 (x_i - b_i)^2$, that is, of non-central $\chi^2$ with $n$ degrees of freedom and non-centrality parameter $(\sum^n_1 b^2_i)^{\frac{1}{2}}.$ Specifically, the distribution of $Q$ can be expressed as a weighted non-central chi-square in the sense that the non-centrality parameter is not fixed but is rather a random variable with a given distribution depending on the $a_i$ and $b_i$.