Generalized likelihood ratio statistics and Wilks phenomenon

Likelihood ratio theory has had tremendous success in parametric inference, due to tile fundamental theory of Wilks. Yet, there is no general applicab le approach for nonparametric inferences based on function estimation. Maxi mum likelihood ratio test statistics in general may not exist in nonparamet ric function estimation setting. Even if they exist, they are hard to find and can not; be optimal as shown in this paper. We introduce the generalize d likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. A new S Wilks phenomenon is unveiled. We demon strate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free a nd follow chi (2)-distributions under null hypotheses for a number of usefu l hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and gen eralized varying coefficient models. We further demonstrate that generalize d likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster. They can even b e adaptively optimal in the sense of Spokoiny by using a simple choice of a daptive smoothing parameter. Our work indicates that the generalized likeli hood ratio statistics are indeed general and powerful for nonparametric tes ting problems based on function estimation.