Some aspects of empirical likelihood

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 devi-ation theory and brie‡y review the history of its development. As a preparation, some of theimportant theorems which we will employ in the following chapters are also introduced.In chapter 3, Asymptotic Optimality of Empirical Likelihood Tests With Weakly DependentData, 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 andidentically distributed data will obey the large deviation principal (LDP) with rate functionequal to the relative entropy, but in general the large deviation performance of empirical measureof dependent data is complicated. In this chapter we add S-mixing condition to the stationaryprocess and we show that the rate function of the LDP of S-mixing process is indeed equal tothe relative entropy, and then asymptotic optimality follows from the large deviation inequality.In chapter 4, Large Deviations of Empirical Likelihood with Nuisance Parameters, we discussthe asymptotic e¢ ciency of empirical likelihood in the presence of nuisance parameters combinedwith augmented moment conditions. We show that in the presence of nuisance parameters, theasymptotic e¢ ciency of the empirical likelihood estimator of the parameter of interest willincrease by adding more moment conditions, in the sense of the positive semide…niteness of thedi¤erence of information matrices. As a by-product, we point out a necessary condition for theasymptotic e¢ ciency to be increased when more moment condition are added. We also deriveasymptotic lower bounds of the minimax risk functions for the estimator of the parameter ofinterest, and we show that the empirical likelihood estimator can achieve this bound.In chapter 5, Empirical Likelihood Estimation of Auction Models via Simulated MomentConditions, we apply empirical likelihood estimation to the simplest …rst-price sealed bid auc-tion model with independent private values. Through estimation of the parameter in the distri-bution function of bidders’private values we consider a potential problem in the EL inferencewhen the moment condition is not in an explicit form and hard to compute, or even not con-tinuous in the parameter of interest. We deal with this issue following the method of simulatedmoment through importance sampling. We demonstrate the convergence of the empirical likeli-hood estimator from the simulated moment condition, and found that the asymptotic varianceis larger than usual which is disturbed by simulation