Empirical Likelihood for Non-Smooth Criterion Functions

Abstract

Suppose that X-1,..., X-n is a sequence of independent random vectors, identically distributed as a d-dimensional random vector X. Let mu is an element of R-p be a parameter of interest and nu is an element of R-q be some nuisance parameter. The unknown, true parameters (mu(0), nu(0)) are uniquely determined by the system of equations E{g(X, mu(0), nu(0))} = 0, where g = (g(1),..., g(p+q)) is a vector of p+q functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter mu(0). 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(1),..., g(p+q) are smooth in mu or nu. 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.