An easy empirical likelihood approach to improved estimation

Shan Wang, Purdue University

Abstract

In this thesis, we construct improved estimates of linear functionals of a probability measure with side information using an easy empirical likelihood (EL) approach. We allow constraint functions which determine side information to grow with the sample size and the use of estimated constraint functions. This is the case in applications to semiparametric models where the constraint functions may depend on the nuisance parameter. We derive the asymptotic normality of the estimates. As applications of the developed results, we construct semiparametrically efficient estimates for statistical depth functions with side information and for linear functionals of a bivariate probability measure with known marginals and with equal but unknown marginals. Another important application is the improved estimation in the structural equation models (SEM). While only information up to second moments is used in SEM, the easy EL- approach provides a convenient procedure to utilize a broad type of side information. As illustration, we study the side information (1) known medians, (2) known third moment, (3) known coefficient of variation, and (4) known correlation coefficient. In SEM, the random errors and random covariates are modeled as uncorrelated as correlation is determined by second order moments. Here we propose to model them to be independent and use the easy EL- approach to utilize the information contained in the independence assumption to improve the efficiency of estimation. A large simulation is conducted to investigate the efficiency gain of the easy EL- approach over the usual approaches.

Degree

Ph.D.

Advisors

Peng, Purdue University.

Subject Area

Mathematics

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