Generalized likelihood ratio statistics and wilks phenomenon

Jianqing Fan, Chunming Zhang, Jian Zhang

Research output: Contribution to journalArticlepeer-review

478 Scopus citations


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 x2-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.

Original languageEnglish (US)
Pages (from-to)153-193
Number of pages41
JournalAnnals of Statistics
Issue number1
StatePublished - Feb 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Asymptotic null distribution
  • Gaussian white noise models
  • Generalized likelihood
  • Nonparametric test
  • Optimal rates
  • Power function
  • Wilks thoerem


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