Generalized likelihood ratio statistics and Wilks phenomenon

Fan, Jianqing and Zhang, Chunming and Zhang, Jian (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Annals of Statistics, 29 (1). pp. 153-193. ISSN 0090-5364. (The full text of this publication is not available from this repository)

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Abstract

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 setting. Even if they exist, they are hard to find and can not; be optimal as shown in this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. A new S 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 given by Ingster. They can even be adaptively optimal in the sense of Spokoiny by using a simple choice of adaptive smoothing parameter. Our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation.

Item Type: Article
Uncontrolled keywords: GOODNESS-OF-FIT; NONPARAMETRIC REGRESSION; ASYMPTOTIC EQUIVALENCE; WHITE-NOISE; MODELS; HYPOTHESIS; TESTS; COEFFICIENT; CHECKING
Subjects: H Social Sciences > HA Statistics
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science
Depositing User: Judith Broom
Date Deposited: 19 Dec 2007 18:22
Last Modified: 07 Apr 2014 10:26
Resource URI: http://kar.kent.ac.uk/id/eprint/594 (The current URI for this page, for reference purposes)
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