Abstract
Maximum likelihood ratio test statistics may not exist in general in nonparametric function estimation setting. In this paper a new class of generalized likelihood ratio (GLR) tests is proposed for nonparametric goodness-of-fit testing via the asymptotic variant of the minimax approach. The proposed nonparametric tests are developed to be asymptotically distribution-free based on latent variable representations. The nonparametric tests are ameliorated to be appropriately complex so that they are analytically tractable and numerically feasible. They are well applicable for the “adaptive” study of hypothesis testing problems of growing dimensions. To assess the proposed GLR tests, the asymptotic properties are derived. The procedure can be viewed as a novel nonparametric extension of the classical parametric likelihood ratio test as a guard against possible gross misspecification of the data-generating mechanism. Simulations of the proposed minimax-type GLR tests are investigated for the small sample size performance and show that the GLR tests have appealing small sample size properties.
Highlights
Parametric models have the advantage of easy interpretation and efficient computation over nonparametric models
In this paper a new class of generalized likelihood ratio (GLR) tests is proposed for nonparametric goodness-of-fit testing via the asymptotic variant of the minimax approach
Simulations of the proposed minimax-type GLR tests are investigated for the small sample size performance and show that the GLR tests have appealing small sample size properties
Summary
Parametric models have the advantage of easy interpretation and efficient computation over nonparametric models. The classical approaches based on L2 and L∞ are popular in nonparametric goodness-of-fit tests They measure the difference between the estimators under null and alternative models and are the generalization of the KolmogorovSmirnov (KS) and Cramer-von Mises (CV) types of statistics. Some likelihood ratio test procedures that are distribution-free under parametric alternatives may become dependent on nuisance parameters under nonparametric alternatives since infinite dimensional neighborhood is around a null hypothesis. To attenuate these difficulties arising from the nonparametric alternatives problems due to the curse of dimensionality, an approach based on the asymptotic variant of the minimax approach is proposed along the line of parametric likelihood ratio tests that possesses distribution-free property. All mathematical proofs of main results are collected in an appendix
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