Distribution of Quadratic Forms and Some Applications

Abstract

The authors were prompted by a general problem concerning hit probabilities arising in military operations to seek the distribution of $Q_i = \sum^k_{i=1}a_ix^2_i, k = 2, 3,$ where the $x_i$ are normally and independently distributed with zero mean and unit variance, $\sum a_i = 1,$ and $a_i > 0.$ While the distribution of a positive definite quadratic form in independent normal variates has been the subject of several papers in recent years [6], [11], [12], laborious computations are required to prepare from existing results the percentiles of the distribution and a table of hit probabilities. This paper discusses the exact distribution of $Q_k$ and then obtains and tabulates the distributions of $Q_2$ and $Q_3,$ accurate to four places. Three other approaches to the distributions are discussed and compared with the exact results: a derivation by Hotelling [8], the Cornish-Fisher asymptotic approximation [3], and the approximation obtained by replacing the quadratic form with a chi-square variate whose first two moments are equated to those of the quadratic form--a type of approximation used in components of variance analysis. The exact values and the approximations are given in Tables I and II. The tables have been prepared with the original problem in mind, but also serve as an aid in several problems arising out of quite different contexts, [1], [2], [13]. These are discussed in Section 6.