A simple four-moment approximation to the distribution of a positive definite quadratic form, with applications to testing

Hillier, Grant and O'Brien, Raymond (2022) A simple four-moment approximation to the distribution of a positive definite quadratic form, with applications to testing (Cemmap Working Paper Series, CWP02/22) London. Centre for Microdata Methods and Practice 29pp. (Submitted)

Record type: Monograph (Discussion Paper)

Abstract

The exact distribution of a quadratic form in n standard normal variables,
Q; say, (or, equivalently, a linear combination of independent chi-squared vari-
ates) is, except in special cases, quite complicated. This has led to many
proposals for approximating the distribution by a more tractable form. These
approximations typically exploit the fact that the cumulants of the distribution
are quite simple, and include both saddlepoint methods, and methods that re-
place the actual statistic with a statistic with the same low-order cumulants
(or moments). In this paper we propose an approximation of this type that
matches the first four moments of the distribution. Its advantage over other
methods is that it is extremely easy to implement, and, as we shall show, it is
almost as accurate as the best of the other proposed methods (which matches
the first eight cumulants). Using the same approach, we also suggest an ap-
proximation to the distribution of the analogue of a regression t - statistic
in cases where the numerator is standard normal, but the denominator is
p
Q,
with Q an independent quadratic form (but not chisquared). This is also shown
to work extremely well.