Enhancements of Non-parametric Generalized Likelihood Ratio Test: Bias Correction and Dimension Reduction

Non-parametric generalized likelihood ratio test is a popular method of model checking for regressions. However, there are two issues that may be the barriers for its powerfulness: existing bias term and curse of dimensionality. The purpose of this paper is thus twofold: a bias reduction is suggested and a dimension reduction-based adaptive-to-model enhancement is recommended to promote the power performance. The proposed test statistic still possesses the Wilks phenomenon and behaves like a test with only one covariate. Thus, it converges to its limit at a much faster rate and is much more sensitive to alternative models than the classical non-parametric generalized likelihood ratio test. As a by-product, we also prove that the bias-corrected test is more efficient than the one without bias reduction in the sense that its asymptotic variance is smaller. Simulation studies and a real data analysis are conducted to evaluate of proposed tests.