Review of Khmaladze Transformations with Their Applications on Testing Theorem
When addressing composite hypothesis testing,empirical process-based statistical tests often lack distribution-free.The Khmaladze transformation,includ-ing both the martingale and unitary transformations,provides an effective solution to this challenge.Initially,in the early 1980s,Khmaladze introduced the martin-gale transformation,specifically designed for testing problems involving continuous distribution functions.Over time,the theoretical foundation of the martingale trans-formation 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 dis-tributions but also to continuous distributions,bringing new perspectives and tools to the field of statistics.However,despite a certain research foundation for the Khmal-adze transformation internationally,these two transformation methods have not yet been fully recognized in China.This article aims to sort out the origin,theoreti-cal principles,development process,and current application status of the Khmaladze transformation and to propose some thoughts on its further development and appli-cation prospects.
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