Fast and accurate calculation for the distributions of Quadratic forms of centered Gaussian variables is of interest in computational statistics . This paper presents a novel numerical procedure to efficiently compute the moments of a given quadratic form. Based on that, a gamma distribution with matched skewness-kurtosis ratio is proposed to approximate its distribution. Comparing with existing methods, the new method is significantly more accurate in getting the right-tail probability. The new method may facilitate the hypothesis testing in the analysis of big data , where efficient and accurate calculation of  small p-values is desired. Relevant R functions are provided in the Supplementary Materials .
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 .
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In computational and applied statistics, it is of great interest to get fast and
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Quadratic forms of centered Gaussian variables is of interest in computational statistics
the quadratic forms of Gaussian random variables
R
numerical procedure to efficiently compute
approximation strategy that contains two developments. First, we propose a faster numerical procedure in computing
R
a given quadratic form. Based on that, a gamma distribution with matched skewness-kurtosis ratio
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)
R
approximate its distribution. Comparing with existing methods, the new method is significantly more accurate in getting the right-tail probability. The new method may facilitate the
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
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the analysis of big data
big data analysis
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very
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Relevant R functions are provided in the Supplementary Materials
An R package Qapprox that implements related methods is available on CRAN