高斯变量二次型分布的高效且精确的近似方法
An efficient and accurate approximation to the distribution of quadratic forms of Gaussian variables
AI总结：
本文提出一种基于矩匹配框架的矩比率方法（MR），用于高效精确近似高斯变量二次型分布，显著提升远右尾概率精度，并适用于大数据分析中的小p值计算。
摘要：
在计算统计学和应用统计学中，快速且精确地计算高斯随机变量二次型的分布具有重要价值。本文提出了一种新颖的近似策略，包含两项进展。首先，我们提出了一种更快的数值程序来计算二次型的矩。其次，我们建立了一个通用的矩匹配框架用于分布近似，该框架涵盖了现有的高斯变量二次型分布的近似方法。在此框架下，提出了一种新的矩比率方法（MR），基于伽马分布匹配偏度和峰度的比率。我们的大量模拟表明：1）MR的精度几乎与精确分布计算相当，且效率更高；2）与现有近似方法相比，MR显著提高了远右尾概率的近似精度。所提出的方法具有广泛的应用。例如，在大数据分析中，当需要高效且精确地计算非常小的p值时，它比现有方法更优。实现相关方法的R包Qapprox可在CRAN上获取。
英文摘要：
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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