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Generalized likelihood ratio statistics have been proposed in Fan, Zhang andZhang [Ann. Statist. 29 (2001) 153-193] as a generally applicable method fortesting nonparametric hypotheses about nonparametric functions. The likelihoodratio statistics are constructed based on the assumption that the distributionsof stochastic errors are in a certain parametric family. We extend their workto the case where the error distribution is completely unspecified via newlyproposed sieve empirical likelihood ratio (SELR) tests. The approach is alsoapplied to test conditional estimating equations on the distributions ofstochastic errors. It is shown that the proposed SELR statistics followasymptotically rescaled \chi^2-distributions, with the scale constants and thedegrees of freedom being independent of the nuisance parameters. Thisdemonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann.Statist. 29 (2001) 153-193] continues to hold under more relaxed models and alarger class of techniques. The asymptotic power of the proposed test is alsoderived, which achieves the optimal rate for nonparametric hypothesis testing.The proposed approach has two advantages over the generalized likelihood ratiomethod: it requires one only to specify some conditional estimating equationsrather than the entire distribution of the stochastic error, and the procedureadapts automatically to the unknown error distribution includingheteroscedasticity. A simulation study is conducted to evaluate our proposedprocedure empirically.

Sieve empirical likelihood ratio tests for nonparametric functions