Distribution of Quadratic Forms and Some Applications

The authors were prompted by a general problem concerning hit probabilities arising in military operations to seek the distribution of Qᵢ = ∑ᵏᵢ₌₁aᵢx²ᵢ, k = 2, 3, where the xᵢ are normally and independently distributed with zero mean and unit variance, ∑ aᵢ = 1, and aᵢ > 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 Qₖ and then obtains and tabulates the distributions of Q₂ and Q₃, 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.