Khmaladze变换及其在检验理论中的应用研究综述  

Review of Khmaladze Transformations with Their Applications on Testing Theorem

作　　者：陈强[1,2] CHEN Qiang(School of Economics,Shanghai University of Finance and Economics,Shanghai 200433,China;Key Laboratory of Mathematical Economics(SUFE),Ministry of Education,Shanghai 200433,China)

机构地区：[1]上海财经大学经济学院,上海200433 [2]上海财经大学数理经济学教育部重点实验室,上海200433

出　　处：《计量经济学报》2024年第4期1031-1063,共33页China Journal of Econometrics

基　　金：中央高校基本科研业务费专项资金(2023110139)

摘　　要：在探讨复合假设检验问题时,基于经验过程的检验统计量往往缺乏分布无关性.Khmaladze变换,包括鞅变换和酉变换,为克服这一难题提供了有效的解决方案.首先,20世纪80年代初期,Khmaladze提出了鞅变换,专门用于处理连续分布函数的检验问题.随着时间的推移,鞅变换的理论基础得到了不断的深化和完善;其应用范围也日益扩大,涵盖了分布函数与回归模型等众多检验问题.进入21世纪,Khmaladze在2013年和2016年进一步提出了酉变换.其不仅适用于离散分布,也适用于连续分布,为统计学领域带来了新的视角和工具.然而,尽管Khmaladze变换在国际上已有一定的研究基础,但在中国,这两种变换方法的研究和应用尚未得到充分的认识.本文旨在对Khmaladze变换的起源、理论原理、发展过程以及当前的应用状况进行梳理,并对其进一步拓展和应用前景提出一些思考.When addressing composite hypothesis testing,empirical process-based statistical tests often lack distribution-free.The Khmaladze transformation,including both the martingale and unitary transformations,provides an effective solution to this challenge.Initially,in the early 1980s,Khmaladze introduced the martingale transformation,specifically designed for testing problems involving continuous distribution functions.Over time,the theoretical foundation of the martingale transformation has been continuously deepened and refined;and its application scope has broadened,covering a wide range of testing problems,including distribution functions and regression models.Entering the 21st century,Khmaladze further proposed the unitary transformation in 2013 and 2016,which is applicable not only to discrete distributions but also to continuous distributions,bringing new perspectives and tools to the field of statistics.However,despite a certain research foundation for the Khmaladze transformation internationally,these two transformation methods have not yet been fully recognized in China.This article aims to sort out the origin,theoretical principles,development process,and current application status of the Khmaladze transformation and to propose some thoughts on its further development and application prospects.

关 键 词：经验过程 检验 分布无关性 鞅变换 酉变换 

分 类 号：O212.1[理学—概率论与数理统计]