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

The exact distribution of a quadratic form in n standard normal variables, Q; say, (or, equivalently, a linear combination of independent chi-squared variates) 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 replace 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 approximation to the distribution of the analogue of a regression t – statistic in cases where the numerator is standard normal, but the denominator is √Q with Q an independent quadratic form (but not chisquared). This is also shown to work extremely well. The approach has applications in many disciplines, from statistics and econometrics through to theoretical physics.