Empirical Likelihood Inference Inequality Constraints

Empirical likelihood is a favorable nonparametric method, and performs bet-ter than traditional nonparametric methods which are based on normal approx-imation. The empirical likelihood approach has many advantages, for example, asymptotic variance is unnecessary to be estimated:the constructed confidence regions are guaranteed to be in parameter space; confidence regions are not pre-determined to be symmetric. Therefore, empirical likelihood has been paid much attention to by many scholars and been applied to many research areas in statis-tics since it was introduced by Owen in 1988.A large amount of theoretical and applied studies on the empirical likeli-hood have been presented in the literature. The obtained results are mostly under regular conditions, i.e., the parameter space is unconstrained and true pa-rameter value is an inner point of parameter space. In such regular situation, Owen showed that the empirical likelihood ratio statistics are asymptotically distributed as chi-square, which is viewed as the Wilks Theorem in nonparame-ter settings. However, parameter space may be constrained in many theoretical studies and practical problems. The common form is inequality constraint. We are often interested in testing whether true parameter value is on the boundary of parameter space. In such irregular situation, how to compute the empirical likelihood ratio statistics and obtain the asymptotical distributions is listed as a challenging problem in Owen[3].In this dissertation, empirical likelihood is applied to population mean model, two-sample problem, linear model and mixed effect model with inequality con-straints. Additionally, asymptotic distributions of the empirical likelihood ratio test statistics are proven to be mixture of chi-square distributions.In Chapter 2, the empirical likelihood for testing problems on population mean with inequality constraints are studied. We consider one-sided and two-sided testing problems, and show that the asymptotic distributions of the em-pirical likelihood ratio test statistics are mixture of chi-square distributions with equal weights. Our proof is new and simple. The local asymptotic power of one-sided testing problem is also explored.In Chapter 3, the empirical likelihood is applied to the two-sample mean problem and two-sample linear regression model with an ordered constraint. The asymptotic distributions of empirical likelihood ratio test statistics are proven to be weighted mixture of chi-square distribution for the two-sample mean problem. For the two-sample linear regression model, the asymptotic distribution of the adjusted empirical likelihood ratio test statistics are proven to be a weighted mixture of chi-square distribution.In Chapter 4, empirical likelihood is applied to the multiple linear regres-sion model and mixed effect model with inequality constraints. For simple linear regression model, the regression coefficient with inequality constraints is studied. We prove that asymptotic distribution of the adjusted empirical likelihood ratio test statistic is a weighted mixture of chi-square distribution. For mixture effect model, we are interested in the existence of the random effect term, i.e., whether the variance of random effect is zero. In this dissertation, the asymptotic distri-bution of the adjusted empirical likelihood ratio test statistic is proven to be a weighted mixture of chi-square distribution.In this dissertation, simulation studies are conducted for our proposed test statistics to evaluate their numerical performances. Numerical results show that our method is more effective. Moreover, when parameter space is constrained, our method is better than conventional approach without considering constraint.