| Maximum likelihood ratio theory contributes tremendous success to parametric inferences, due to the fundamental theory of Wilks (1938). However, there is no generally applicable approach for inferences based on nonparametric function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to find and can not be optimal as shown in this work. The purpose of this dissertation is to propose a generally accepted technique for hypothesis testing. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks' phenomenon is discovered. We demonstrate that the generalized likelihood statistics are asymptotically distribution free and nearly follow chi2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models. Our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric inferences based on function estimation. Issues on related methodologies and applications will be also discussed. |
