The Annals of Mathematical Statistics

Approximations to the Distribution of Quadratic Forms

M. M. Siddiqui

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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.

Article information

Source
Ann. Math. Statist., Volume 36, Number 2 (1965), 677-682.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177700175

Digital Object Identifier
doi:10.1214/aoms/1177700175

Mathematical Reviews number (MathSciNet)
MR177467

Zentralblatt MATH identifier
0136.41202

JSTOR
links.jstor.org

Citation

Siddiqui, M. M. Approximations to the Distribution of Quadratic Forms. Ann. Math. Statist. 36 (1965), no. 2, 677--682. doi:10.1214/aoms/1177700175. https://projecteuclid.org/euclid.aoms/1177700175


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