The Annals of Statistics

Sieve empirical likelihood ratio tests for nonparametric functions

Jianqing Fan and Jian Zhang

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Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the distributions of stochastic errors. It is shown that the proposed SELR statistics follow asymptotically rescaled χ2-distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193] continues to hold under more relaxed models and a larger class of techniques. The asymptotic power of the proposed test is also derived, which achieves the optimal rate for nonparametric hypothesis testing. The proposed approach has two advantages over the generalized likelihood ratio method: it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error, and the procedure adapts automatically to the unknown error distribution including heteroscedasticity. A simulation study is conducted to evaluate our proposed procedure empirically.

Article information

Ann. Statist., Volume 32, Number 5 (2004), 1858-1907.

First available in Project Euclid: 27 October 2004

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G10 62J12

Nonparametric test sieve empirical likelihood conditional estimating equations Wilks’ theorem varying coefficient models


Fan, Jianqing; Zhang, Jian. Sieve empirical likelihood ratio tests for nonparametric functions. Ann. Statist. 32 (2004), no. 5, 1858--1907. doi:10.1214/009053604000000210.

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