A New Result on the Distribution of Quadratic Forms

The distribution of the non-homogeneous quadratic form Q = ⁿ₁ aᵢ (xᵢ - bᵢ) ², where the xᵢ are independent standardized normal variables and the aᵢ and bᵢ are real constants with aᵢ > 0, has recently 1 been obtained as an infinite linear combination in scaled central and noncentral ² distribution functions in the form equation*1. 1P Q t = ^₀ cⱼ (p) F₍+₂₉ (t/p) = ^₀ dⱼ (p) G₍+₂₉;₊ (t/p). equation* Here p is an arbitrary positive constant, F₍ + ₂₉ () is the distribution function of ² with n + 2j degrees of freedom and G₍ + ₂₉;₊ () is the distribution function of ² with n + 2j degrees of freedom and non-centrality parameter = (ⁿ₁ b²ᵢ) ^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 ᵢaᵢ 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 cⱼ (p) will be expressed in simpler and more appealing form than in 1. In essence, the distribution of Q (including that of ⁿ₁ aᵢx²ᵢ) is found to be almost a direct consequence of the distribution of the special non-homogeneous form ⁿ₁ (xᵢ - bᵢ) ², that is, of non-central ² with n degrees of freedom and non-centrality parameter (ⁿ₁ b²ᵢ) ^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 aᵢ and bᵢ.