  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 \chi^2-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.
