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

Abstract

Nonparametric generalized likelihood ratio test is popularly used for model checking for regressions. However, there are two issues that may be the barriers for its powerfulness. First, the bias term in its liming null distribution causes the test not to well control type I error and thus Monte Carlo approximation for critical value determination is required. Second, it severely suffers from the curse of dimensionality due to the use of multivariate nonparametric function estimation.The purpose of this paper is thus two-fold: a bias-correction is suggested to this test and a dimension reduction-based model-adaptive enhancement is recommended to promote the power performance. The proposed test still possesses the Wilks phenomenon, and the test statistic can converge to its limit at a much faster rate and is much more sensitive to alternative models than the original nonparametric generalized likelihood ratio test as if the dimension of covariates were one. Simulation studies are conducted to evaluate the finite sample performance and to compare with other popularly used tests. A real data analysis is conducted for illustration.

Model checking for parametric single‐index quantile models

“…In this work, we aim to perform model testing in quantile regression, and we use sufficient dimension reduction (SDR) to circumvent the curse of dimensionality in multivariate nonparametric estimation. For the application of SDR in mean regression, we refer the reader to Guo, Wang, and Zhu (2016), Niu, Guo, and Zhu (2018), and Tan and Zhu (2019) for more details. Readers can refer to an overview of model checking under the mean regression framework in Guo and Zhu (2017).…”