二次型统计量与柯西统计量  

Quadratic and Cauchy Statistics

作　　者：王子轩 王晋娟 李启寨[1,2] WANG Zi-xuan;WANG Jin-juan;LI Qi-zhai(Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China;School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China;School of Mathematics and Statistics,Beijing Institute of Technology,Beijing 100081,China)

机构地区：[1]中国科学院数学与系统科学研究院,北京100190 [2]中国科学院大学数学科学学院,北京100049 [3]北京理工大学数学与统计学院,北京100081

出　　处：《数理统计与管理》2021年第3期405-416,共12页Journal of Applied Statistics and Management

基　　金：国家自然科学基金(11722113);北京市自然科学基金(RZ180006).

摘　　要：二次型统计量是多元统计分析中假设检验问题的常用统计量,许多假设检验问题的统计量都可以转换成二次型统计量的形式.在零假设成立的条件下,二次型统计量渐近服从卡方分布.本文通过对协方差矩阵进行特征值分解,构造了一族二次型统计量,它包含常见的二次型统计量.在零假设成立的条件下,我们证明了族中每个二次型统计量的渐近分布与若干个相互独立服从自由度为1的卡方分布的随机变量加权和相同.由于计算其统计显著性较为困难,我们提出利用广义伽马分布来近似统计量的渐近分布,进而,我们构造了一个柯西统计量并给出计算其统计显著性的显式表达式.模拟结果和实例分析表明,柯西统计量比二次型统计量更稳健,许多情况下功效更高.Many test statistics in multivariate analysis can be transformed into quadratic statistics.Under the null hypothesis,the quadratic statistics obey Chi-square distribution asymptotically.In this paper,we propose a family of quadratic statistics via the eigenvalue decomposition of the covariance matrix.Under the null hypothesis,we prove that each quadratic statistic in the family is asymptotically equivalent to the weighted sum of several random variables that are independent and obey Chi-square distribution with one degree-of-freedom.Because it is difficult to calculate its statistical significance,we propose to use the generalized gamma distribution to approximate the asymptotic distribution and then construct a Cauchy statistic whose statistical significance is provided explicitly.Simulation results and a real data analysis show that the Cauchy statistics is more robust and efficient than the quadratic statistics.

关 键 词：二次型统计量 柯西统计量 广义伽马分布 柯西分布 

分 类 号：O212[理学—概率论与数理统计]