对数线性模型下基于φ-散度的单边检验  

One-sided Test Based on φ-divergence under Log-linear Models

作　　者：金应华[1] 吴耀华[2] 邵全喜 JIN YINGHUA;WU YAOHUA;SHAO QUANXI(School of Applied Mathematics,Guangdong University of Technology,Guangzhou 510006,China;Department of Statistics and Finance,University of Science and Technology of China,Hefei 230026,China;CSIRO Data61,Leeuwin Centre,65 Broockway Road,Floreat,WA 6014,Australi)

机构地区：[1]广东工业大学应用数学学院,广州510006 [2]中国科学技术大学统计与金融系,合肥230026 [3]CSIRO Data61, Leeuwin Centre, 65 Broockway Road, Floreat, WA 6014, Australia

出　　处：《应用数学学报》2018年第4期497-510,共14页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11401114);广东省自然科学基金(S2012040007622)资助项目

摘　　要：单边检验是假设检验理论的重要组成部分之一.本文研究了在乘积多项抽样对数线性模型下的某种单边假设检验问题.基于Ф-散度和约束最小Ф散度估计（RMФE）,提出了三类检验统计量,证明了它们有相同的渐近分布,即类卡方（Chi-bar-square）分布.此三类统计量包含似然比（likelihood ratio）统计量和皮尔逊（Pearson）统计量等特例,推广了现有文献的研究结果.实例分析展示了此三类统计量的检验效果.模拟研究表明,在样本量较小时,功效散度（power-divergence）族中存在比似然比统计量和皮尔逊统计量表现更好的替代.One-sided test is an important part of hypothesis test theory. Some kind of onesided test for log-linear models under product-multinomial sampling is studied in this paper.Based on Ф-divergence and the restricted minimum φ-divergence estimator（RMФDE）, three families of statistic are proposed. All the three proposed statistics have identical asymptotic distribution as Chi-bar-square distribution, and include the likelihood ratio statistic and the Pearson statistic as special cases. These proposed statistics can be treated as a generalization of the results in existing literature with three improvements： the model in this paper is much wider than the saturated log-linear model they considered; the form of one-sided test is much more general than the likelihood ratio ordering test they constructed; and RMφDE used to construct these statistic is more general than the maximum likelihood estimator（MLE） they used. Real data analysis demonstrates the effect of these statistic. A simulation study shows that some members of the power-divergence family could act as superior alternatives to the likelihood ratio statistic and the Pearson statistic under finite sample sizes.

关 键 词：单边检验 Ф-散度 乘积多项抽样 对数线性模型 类卡方分布 

分 类 号：O212.1[理学—概率论与数理统计]