Empirical Likelihood for Non-Smooth Criterion Functions

摘要

Suppose that X_1,...,X_n is a sequence of independent random vectors, identically distributed as a d-dimensional random vector X. Let μ∈R~p be a parameter of interest and v∈R~q be some nuisance parameter. The unknown, true parameters (μ_0, v_0) are uniquely determined by the system of equations E{g(X, μ_0, v_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 μ_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 μ or v. 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.