Energy distance的配对样本分布差异检验法
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摘要
讨论了配对样本分布差异的检验问题,基于Energy distance的概念构造出了一种新的检验配对样本分布差异的检验统计量,该统计量具有U统计量形式,基于U统计量理论讨论了该检验统计量的大样本性质,导出了该检验统计量在零假设即配对变量同分布的情况下的渐近分布,最后采用bootstrap重抽样技术进行数值模拟说明该方法的有效性.本文提出的检验方法适用于任意有限维的具有有限一阶矩的随机向量.
A new test based on Energy distance is proposed for detecting the equality of distribution for paired samples. The test statistic has the form of a U statistic. Asymptotic null distribution of this statistic is derived based on the theory of U statistic. A simulation via bootstrap method shows that our test statistic has good performance for several situations. Our test method is suitable for arbitrary dimensional random variables with finite first moment.
A new test based on Energy distance is proposed for detecting the equality of distribution for paired samples. The test statistic has the form of a U statistic. Asymptotic null distribution of this statistic is derived based on the theory of U statistic. A simulation via bootstrap method shows that our test statistic has good performance for several situations. Our test method is suitable for arbitrary dimensional random variables with finite first moment.
引文
[1] Székely G J. E-Statistics:the energy of statistical samples[R]. Bowling Green State University,Department of Mathematics and Statistics Technical Report,2003,(3-5):2000~2003.
[2] Rizzo M L. A test of homogeneity for two multivariate populations[R]. Proceedings of the American Statistical Association,Physical and Engineering Sciences Section,2002.
[3] Székely G J,Rizzo M L. A new test for multivariate normality[J]. Journal of Multivariate Analysis,2005,93(1):58~80.
[4] Székely G J,Rizzo M L. Hierarchical clustering via joint between-within distances:extending Ward's minimum variance method[J]. Journal of classification,2005,22(2):151~183.
[5] Székely G J,Rizzo M L,Bakirov N K. Measuring and testing dependence by correlation of distances[J]. The annals of statistics,2007,35(6):2769~2794.
[6] Rizzo M L,Székely G J. Disco analysis:a nonparametric extension of analysis of variance[J]. The Annals of Applied Statistics,2010,4(2):1034~1055.
[7] Székely G J,Rizzo M L. Energy statistics:a class of statistics based on distances[J]. Journal of statistical planning and inference,2013,143(8):1249~1272.
[8]安鸿志.关于绝对矩的几个性质[J].应用概率统计,2006,22(3):233~236.
[9]王芳,崔恒建,金蛟.分布对称中心的U检验方法[J].北京师范大学学报(自然科学版),2009,(1):17~21.
[10] Lee A J. U-Statistics:theory and practice[M]. Statistics:Textbooks and Monographs M Dekker,1990.
[2] Rizzo M L. A test of homogeneity for two multivariate populations[R]. Proceedings of the American Statistical Association,Physical and Engineering Sciences Section,2002.
[3] Székely G J,Rizzo M L. A new test for multivariate normality[J]. Journal of Multivariate Analysis,2005,93(1):58~80.
[4] Székely G J,Rizzo M L. Hierarchical clustering via joint between-within distances:extending Ward's minimum variance method[J]. Journal of classification,2005,22(2):151~183.
[5] Székely G J,Rizzo M L,Bakirov N K. Measuring and testing dependence by correlation of distances[J]. The annals of statistics,2007,35(6):2769~2794.
[6] Rizzo M L,Székely G J. Disco analysis:a nonparametric extension of analysis of variance[J]. The Annals of Applied Statistics,2010,4(2):1034~1055.
[7] Székely G J,Rizzo M L. Energy statistics:a class of statistics based on distances[J]. Journal of statistical planning and inference,2013,143(8):1249~1272.
[8]安鸿志.关于绝对矩的几个性质[J].应用概率统计,2006,22(3):233~236.
[9]王芳,崔恒建,金蛟.分布对称中心的U检验方法[J].北京师范大学学报(自然科学版),2009,(1):17~21.
[10] Lee A J. U-Statistics:theory and practice[M]. Statistics:Textbooks and Monographs M Dekker,1990.