Approximations to the Distribution of Quadratic Forms

Abstract

Positive definite quadratic forms in normal variates, which do not necessarily reduce to a multiple of a $\chi^2$ variate, arise quite naturally in estimation and hypothesis testing problems related to normal distributions and processes. A classical example is the problem of testing the difference between two sample means, $\bar x - \bar y$, where the $x$ observations have variance other than of $y$ observations (Welch [9]). More recent examples include: analysis of variance when errors are assumed to have unequal variance or are correlated (Box [1] [2]); regression analysis with stationary errors (Siddiqui [8]); and estimation of spectral density functions of stationary processes (Freiberger and Grenander [4]). Let $Q = \frac{1}{2}Y'MY$, where $Y = \lbrack Y_1, \cdots, Y_n\rbrack$ is a $N(0; V)$ distributed column vector, $Y'$ its transposed row vector, 0 zero vector, $V$ a positive definite covariance matrix, and $M$ a real symmetric matrix of rank $m \geqq n$. Let $a_1, \cdots, a_m$ be the non-zero characteristic roots of $A = MV$. It is well known (see, for example, Ruben [8]) that there exists a non-singular transformation from $Y$ to $X$ such that $X_1, \cdots, X_n$ are independent $N(0, 1)$ variates and $Q = \frac{1}{2} \sum^m_1 a_jX^2_j$. Without loss of generality we therefore assume that $Q$ has this canonical form. Many papers have been written on the distribution of $Q$, especially when $a_j$ are positive, and a more or less comprehensive list of these is included in the references of the two papers by Ruben [8] [9]. We shall therefore refer to only those which have direct bearing with the present paper. In this paper, we will be mainly concerned with distribution of $Q$ when $m$ is an even number, say $2k$, and $a_j$ positive. When $m$ is an odd number a slight modification is necessary and this is mentioned in Remark (2) of Section 3. We will choose our subscripts so that $0 < a_1 \leqq a_2 \cdots \leqq a_{2k}$. After some notation and preliminaries in Section 2, a well known result will be stated as Theorem 1 under which $F(x) = Pr (Q > x)$ can be evaluated as a finite linear combination of gamma df's. In other situations we require some methods of approximating to $F(x)$. In Sections 3 and 4 a simple approximation to $F(x)$ will be presented which reduces to the exact distribution when the condition of Theorem 1 is satisfied. The method is based on bounding $Q$ by $Q_1$ and $Q_2$, where $Q_1$ and $Q_2$ are quadratic forms satisfying the condition of Theorem 1. The approximation is then obtained by minimizing $d(F, \hat F)$ where $\hat F(x)$ is a linear combination of $F_i(x) = Pr (Q_i > x), i = 1, 2$, and $d(\cdot, \cdot)$ is the distance function of the metric space $L^2(0, \infty)$. In Section 5 a few numerical examples will be worked out for purposes of illustration.