Regression models with censored data including censored linear regression model, censored partial linear models and the Cox regression model have wide applications in many fields. In this thesis we discuss several problems in these models, First, we investigate the empirical likelihood method and Edgeworth expansion in censored linear regression model. An empirical likelihood ratio for the slope parameter vector of the model is defined and shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the situation when empirical likelihood applies to the linear regression model with no censoring. We also establish a one-term Edgeworth expansion for the Koul, Susarla and Van Ryzin type estimator of the slope parameter, our approach is to...[ Read more ]
Regression models with censored data including censored linear regression model, censored partial linear models and the Cox regression model have wide applications in many fields. In this thesis we discuss several problems in these models, First, we investigate the empirical likelihood method and Edgeworth expansion in censored linear regression model. An empirical likelihood ratio for the slope parameter vector of the model is defined and shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the situation when empirical likelihood applies to the linear regression model with no censoring. We also establish a one-term Edgeworth expansion for the Koul, Susarla and Van Ryzin type estimator of the slope parameter, our approach is to represent the estimator as an asymptotic U-statistic plus some negligible terms and hence apply the known results on the Edgeworth expansions for asymptotic U-statistics. The counting process and martingale techniques are used to provide the proof of the main results. Secondly, we study the empirical likelihood method and the estimation theory in censored partial linear models. An empirical likelihood ratio for the parametric component of the model is defined and shown that its asymptotic distribution is a weighted sum of independent chi-square distributions. The least squares estimator and kernel regression estimator for the parametric component and nonparametric component of the model are proposed respectively, their asymptotic properties, such as the asymptotic normality, convergence rates, Berry-Esseen bound for the convergence rate to normality, are obtained under appropriate conditions. Finally, we investigate the empirical likelihood method for the Cox regression model when the failure times are subject to random censoring. An empirical likelihood ratio for the vector of regression coeficients is defined and shown that its limiting distribution is a chi-square distribution with p degrees of freedom. Some simulation studies are presented to compare the empirical likelihood methods with the normal approximation methods.
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