Some aspects of empirical likelihood

Abstract

Chapter 1 is a non technical introduction to the thesis. In chapter 2, Basics of Large Deviation Theory, we illustrate the basic idea of large deviation theory and briefly review the history of its development. As a preparation, some of the important theorems which we will employ in the following chapters are also introduced.
In chapter 3, Asymptotic Optimality of Empirical Likelihood Tests With Weakly Dependent Data, we extend the result of Kitamura (2001) to stationary mixing data. The key thing inproving the large deviation optimality is that the empirical measure of the independently and identically distributed data will obey the large deviation principal (LDP) with rate function equal to the relative entropy, but in general the large deviation performance of empirical measure of dependent data is complicated. In this chapter we add S-mixing condition to the stationary process and we show that the rate function of the LDP of S-mixing process is indeed equal to the relative entropy, and then asymptotic optimality follows from the large deviation inequality.
In chapter 4, Large Deviations of Empirical Likelihood with Nuisance Parameters, we discuss the asymptotic efficiency of empirical likelihood in the presence of nuisance parameters combined with augmented moment conditions. We show that in the presence of nuisance parameters, the asymptotic efficiency of the empirical likelihood estimator of the parameter of interest will increase by adding more moment conditions, in the sense of the positive semidefiniteness of the difference of information matrices. As a by-product, we point out a necessary condition for the asymptotic efficiency to be increased when more moment condition are added. We also derive asymptotic lower bounds of the minimax risk functions for the estimator of the parameter of interest, and we show that the empirical likelihood estimator can achieve this bound.
In chapter 5, Empirical Likelihood Estimation of Auction Models via Simulated Moment Conditions, we apply empirical likelihood estimation to the simplest first-price sealed bid auction model with independent private values. Through estimation of the parameter in the distribution function of bidders’ private values we consider a potential problem in the EL inference when the moment condition is not in an explicit form and hard to compute, or even not continuous in the parameter of interest. We deal with this issue following the method of simulated moment through importance sampling. We demonstrate the convergence of the empirical likelihood estimator from the simulated moment condition, and found that the asymptotic variance is larger than usual which is disturbed by simulation.

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ORCID for Grant Hillier: orcid.org/0000-0003-3261-5766

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