A Fast and Accurate Approximation to the Distributions of Quadratic Forms of Gaussian Variables
A Fast and Accurate Approximation to the Distributions of Quadratic Forms of Gaussian Variables
复制标题
高斯变量二次型分布的快速准确逼近
DOI：
10.1080/10618600.2021.2000423
复制
发表时间：
2022
影响因子：
2.4
通讯作者：
Wu, Zheyang
中科院分区：
文献类型：
--
作者：
Zhang, Hong;Shen, Judong;Wu, Zheyang
在计算统计和应用统计中，快速准确地计算高斯型随机变量的二次型分布是一个非常重要的问题。本文提出了一种新的近似策略，包含两个发展。首先，我们提出了一个快速的数值计算的二次型的时刻。其次，我们建立了一个通用的矩匹配框架的分布近似，其中包括现有的近似方法的高斯变量的二次型的分布。在此框架下，提出了一种新的矩比方法（MR），以匹配基于伽玛分布的偏度和峰度的比例。我们广泛的模拟表明：（i）MR几乎是准确的精确分布计算，是更快;（ii）与现有的近似方法相比，MR显着提高了近似的精度最右尾概率。该方法具有广泛的应用前景。例如，它是比现有方法更好的选择，用于促进大数据分析中的假设检验，其中需要快速准确地计算非常小的p值。CRAN上有一个实现相关方法的R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 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 smallp-values are desired. An R packageQapproxthat implements related methods is available on CRAN.