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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 article presents a novel approximation strategy that contains two developments. First, we propose a fast 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 (i) MR is almost as accurate as the exact distribution calculation and is much faster; (ii) 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 fast and accurate calculation of very small  p -values are desired. An R package  Qapprox  that implements related methods is available on CRAN.

A Fast and Accurate Approximation to the Distributions of Quadratic Forms of Gaussian Variables