论文标题
高斯变量的二次形式的分布有效，准确的近似
An efficient and accurate approximation to the distribution of quadratic forms of Gaussian variables
论文作者
论文摘要
在计算和应用统计中，对于高斯随机变量的二次形式的分布进行快速准确的计算是非常有趣的。本文提出了一种新的近似策略，其中包含两个发展。首先，我们在计算二次形式的力矩时提出了更快的数值程序。其次，我们建立了一个用于分布近似的一般矩匹配框架，该框架涵盖了高斯变量的二次形式的分布的现有近似方法。在此框架下，提出了一种新颖的力矩比方法（MR），以匹配基于伽马分布的偏度和峰度的比率。我们的广泛模拟表明，1）MR几乎与确切的分布计算一样准确，并且效率更高。 2）与现有的近似方法进行比较，MR显着提高了近似右尾概率的准确性。提出的方法具有广泛的应用。例如，在大数据分析中促进假设检验的现有方法是一个更好的选择，在大数据分析中，需要对非常小的$ p $值进行有效，准确的计算。 CRAN上可以使用实现相关方法的R软件包Qapprox。
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.
