Comparison of parametric and empirical likelihood functions

Abstract

SUMMARY In this paper, parametric and empirical likelihood functions or surfaces are compared. In particular, first- and second-order expansions for log likelihood functions are developed in nonparametric and parametric situations, where the functional of interest is a smooth function of vector means. The two approaches are compared and the precise order of difference between the nonparametric and parametric log likelihood functions is obtained. In general, the surfaces need not agree to first order. However, in certain exponential family models, the surfaces are quite close in a distributional sense. Moreover, empirical likelihood can be thought of as an approximation to a true parametric likelihood based on a parametric least favourable family.