Symmetric Location Estimation/Testing by Empirical Likelihood

Abstract

The problem of estimating the center of a symmetric distribution is well studied and many nonparametric procedures are available. It often serves as the test problem for many nonparametric estimation procedures, and stimulated the development of efficient nonparametric estimation theory. We use this familiar setting to illustrate a novel use of empirical likelihood method for estimation and testing. Empirical likelihood is a general nonparametric inference method, see Owen [Owen, A. (2001). Empirical Likelihood. London: Chapman and Hall]. However, for symmetric location problem (and some other problems) empirical likelihood has difficulties. Owen (2001) call them “challenges for the empirical likelihood”. We propose and study a way to use the empirical likelihood with such problems by modifying the parameter space. We illustrate this approach by applying it to the symmetric location problem. We show that the usual asymptotic theory of empirical likelihood still holds and the asymptotic efficiency of the so obtained empirical NPMLE of location is studied.

Key Words:

• Many constraints of symmetry
• Asymptotic chi-square distribution
• Nonparametric information bound

AMS 1991 Subject Classification:

• Primary 62G10
• Secondary 62G05

Acknowledgments