Empirical Likelihood for Non-Smooth Criterion Functions

Suppose that "X"_<b>1</b>,…, "X"_<b>""n""</b> is a sequence of independent random vectors, identically distributed as a "d"-dimensional random vector "X". Let <formula format="inline"><file name="sjos_640_mu1.gif" type="gif" /></formula> be a parameter of interest and <formula format="inline"><file name="sjos_640_mu2.gif" type="gif" /></formula> be some nuisance parameter. The unknown, true parameters ("μ"_<b>0</b>,"ν"_<b>0</b>) are uniquely determined by the system of equations "E"&lcub;"g"("X","μ"_<b>0</b>,"ν"_<b>0</b>)&rcub ; &equals;  <b>0</b>, where "g" &equals;  ("g"_<b>1</b>,…,"g"<su b><b>""p""</b>&plus;<b>""q""</b></sub>) is a vector of "p"&plus;"q" functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter "μ"_<b>0</b>. The results in this paper are valid under very mild conditions on the vector of criterion functions "g". In particular, we do not require that "g"_<b>1</b>,…,"g"<sub><b>""p""</b>&plus;<b>""q""</b></su b> are smooth in "μ" or "ν". This offers the advantage that the criterion function may involve indicators, which are encountered when considering, e.g. differences of quantiles, copulas, ROC curves, to mention just a few examples. We prove the asymptotic limit of the empirical log-likelihood ratio, and carry out a small simulation study to test the performance of the proposed EL method for small samples. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.