TRUNCATION ERROR IN THE EXPANSION OF THE DISTRIBUTION OF A QUADRATIC FORM

Positive definite quadratic forms in normal variables arise in the problems of estimation and testing of hypotheses relating to the normal distribution. Various formulas are known for expanding the cumulative distribution function and the probability density function of the quadratic form in a series. Johnson and Kotz (1969) have considered an expansion, based on Laguerre polynomials. In this paper we consider the expansion in a series of chi-square density (distribution) functions. We derive bounds on the truncation error in the given expansion. The given bounds are smaller than the corresponding bounds in the expansion based on the Laguerre polynomials. Empirical results are given on the truncation errors in the proposed expansion.