Empirical Likelihood for Non-Smooth Criterion Functions

Suppose that X1,..., Xn is a sequence of independent random vectors, identically distributed as a d-dimensional random vector X. Let be a parameter of interest and be some nuisance parameter. The unknown, true parameters (mu0,nu0) are uniquely determined by the system of equations E{g(X,mu0,nu0)} = 0, where g = (g1,...,gp+q) is a vector of p+q functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter mu0. 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 g1,...,gp+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.

Empirical Likelihood for Non-Smooth Criterion Functions Articles