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Generalized likelihood ratio statistics and Wilks phenomenon. (English) Zbl 1029.62042
Summary: Likelihood ratio theory has had tremendous success in parametric inference, due to the fundamental theory of Wilks. Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation settings. Even if they exist, they are hard to find and can not be optimal as shown in this paper.
We introduce generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. A new Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow $$\chi^2$$-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 and generalized varying coefficient models.
We further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence. They can even be adaptively optimal by using a simple choice of adaptive smoothing parameters. Our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation.

##### MSC:
 62G10 Nonparametric hypothesis testing 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62J12 Generalized linear models (logistic models)
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