Nonparametric And Semiparametric Likelihood Inference Using Auxiliary Information

This paper consists of five chapters.The introduction is located in Chapter1.In Chapter2, we consider a semiparametric model, in which the conditionaldensity of the response given covariates is correctly specified but the marginaldistribution of the covariates is unknown. Responses may be missing. By combiningempirical and parametric likelihoods, we show that the combined semiparametriclikelihood can produce valid inferences for the underlying parameters under themissing-at-random assumption. Based on the combined semiparametric likelihoodfunctions, we also develop Wilks’ type tests and corresponding confidence regionsfor the model parameter and the mean response.In Chapter3, we consider the inference of conditional moment models. Forsuch models, several estimators that achieve the semiparametric efciency boundhave been proposed. However, in many studies, auxiliary information is availableas unconditional moment restrictions. Meanwhile, we also consider the presence ofmissing responses. We propose the combined empirical likelihood (CEL) estimatorto incorporate such auxiliary information to improve the estimation efciency ofthe conditional moment restriction models. We show that, when assuming respons-es are missing at random, the CEL estimator achieves better efciency than theprevious estimators due to utilization of the auxiliary information. Based on theCEL, we also develop Wilks’ type tests and corresponding confidence regions forthe model parameter and the mean response. Since kernel smoothing is used, theCEL method may have difculty for problems with high dimensional covariates.In such situations, we propose an instrumental variable-based empirical likelihood(IVEL) method to handle this problem. The merit of the CEL and IVEL arefurther illustrated through simulation studies. In Chapter4, we propose a nonparametric method, called rank-based empir-ical likelihood (REL), for making inferences on medians and cumulative distribu-tion functions (CDFs) of k populations. The standard distribution-free approachto testing the equality of k medians requires that the k population distributionshave the same shape. Our REL-ratio (RELR) test for this problem requires fewerassumptions and can efectively use the symmetry information when the distribu-tions are symmetric. Furthermore, our RELR statistic does not require estimationof variance, and achieves asymptotic pivotalness implicitly. When the k popula-tions have equal medians, we show that the REL method produces valid inferencesfor the common median and CDFs of k populations. Simulation results show thatthe REL approach works remarkably well in finite samples. A real data example isused to illustrate the proposed REL method.In Chapter5, we consider the empirical likelihood inference for censored quan-tile regression model with longitudinal studies.