Some aspects of empirical likelihood.
University of Southampton, Scool of Social Sciences,
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 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 in
proving 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 e¢ ciency 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 e¢ ciency of the empirical likelihood estimator of the parameter of interest will
increase by adding more moment conditions, in the sense of the positive semide…niteness of the
di¤erence of information matrices. As a by-product, we point out a necessary condition for the
asymptotic e¢ ciency 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 …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 inference
when 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 simulated
moment through importance sampling. We demonstrate the convergence of the empirical likeli-
hood 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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