  In computational and applied statistics, it is of great interest to get fast
and accurate calculation for the distributions of the quadratic forms of
Gaussian random variables. This paper presents a novel approximation strategy
that contains two developments. First, we propose a faster numerical procedure
in computing the moments of the quadratic forms. Second, we establish a general
moment-matching framework for distribution approximation, which covers existing
approximation methods for the distributions of the quadratic forms of Gaussian
variables. Under this framework, a novel moment-ratio method (MR) is proposed
to match the ratio of skewness and kurtosis based on the gamma distribution.
Our extensive simulations show that 1) MR is almost as accurate as the exact
distribution calculation and is much more efficient; 2) comparing with existing
approximation methods, MR significantly improves the accuracy of approximating
far right tail probabilities. The proposed method has wide applications. For
example, it is a better choice than existing methods for facilitating
hypothesis testing in big data analysis, where efficient and accurate
calculation of very small $p$-values is desired. An R package Qapprox that
implements related methods is available on CRAN.
