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.

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