Revisiting the generalized likelihood ratio test and its related nonparametric regression inference problems

Nonparametric/semiparametric modelling is a flexible modelling approach widely used in mod- ern data analysis. Nonparametric/semiparametric regression explores the underlying pattern of the relationships between variables. Although the nonparametric/semiparametric regression comes with relatively small bias, it involves paying a price on the variance side. To avoid the price paid on the variance side, a parametric model should be used if one indeed holds. Determining whether a parametric model is suitable requires conducting a generalized likelihood ratio test for a specific parametric or shape constrained nonparametric model vs nonparametric model. The generalized likelihood ratio test has been investigated extensively in the last 15 years. However, some hard problems remain. The main objective of this proposal is to reinvestigate the generalized likelihood ratio test to address some of hard the problems associated with the statistical inference of nonpara- metric/semiparametric regression models. We expect that the developed methodology and theories will apply to other classical nonparametric/semiparametric inference approaches, such as empirical likelihood and discrepancy methods.

The generalized likelihood ratio test statistic is a kind of profile likelihood statistic. It assumes that the error in the regression model follows normal distribution or some special distribution. First, acknowledging other nonparametric testing approaches, we propose a simple way to construct a new generalized likelihood ratio test statistic which is completely nonparametric and robust to dif- ferent regression error distributions. Second, we consider the effect of the bias of the nonparametric regression estimate on the generalized likelihood ratio test statistic. We propose a bias reduction nonparametric regression estimate and apply it to construct the real generalized likelihood ratio test statistic. We also apply the estimate to construct simultaneously confidence bands for the nonpara- metric regression functions. Third, in many statistical inference problems, given prior information or restrictive conditions related to nonparametric functions, the model under the null hypothesis is not a parametric form, but rather a nonparametric form with some restrictive conditions, or the alternative hypothesis space of the model is not a big function space, but rather a relative small function space. We investigate the properties of the generalized likelihood ratio test statistic for those testing problems and apply them to real data analysis. In practice, most samples are often taken from different populations. Studying the properties of the generalized likelihood ratio test statistic for two or more samples inference problems would have wide applications. This represents the fourth objective of this proposal. The bias effect in two or more sample nonparametric testing problems is also an interesting topic to investigate. It is well known that the optimal bandwidth of the nonparametric testing problem is different from that that of the estimation problem. We will investigate how select an efficient bandwidth for the generalized likelihood ratio test statis- tic based on the power function, and how to use the idea of the adaptive Neyman test statistic to avoid the bandwidth selection problem for nonparametric inferences. This is our fifth objec- tive. High-dimensional nonparametric or semiparametric models would be an efficient approach to making statistical inferences for high-dimensional data without much model bias. It is unknown whether Wilk’s phenomenon remains satisfied for the generalized likelihood ratio test statistic in a high-dimensional setting. Our final objective of this proposal is to investigate the properties of the generalized likelihood ratio test statistic for statistical inference problems in high dimensional nonparametric or semiparametric regression models.