The Annals of Statistics

Generalized Likelihood Ratio Statistics and Wilks Phenomenon

Jianqing Fan, Chunming Zhang, and Jian Zhang

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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 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 χ2-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.

Article information

Ann. Statist., Volume 29, Number 1 (2001), 153-193.

First available in Project Euclid: 5 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G10: Hypothesis testing 62J12: Generalized linear models

asymptotic null distribution Gaussian white noise models nonparametric test optimal rates power function generalized likelihood Wilks theorem


Fan, Jianqing; Zhang, Chunming; Zhang, Jian. Generalized Likelihood Ratio Statistics and Wilks Phenomenon. Ann. Statist. 29 (2001), no. 1, 153--193. doi:10.1214/aos/996986505.

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