Empirical Likelihood And Penalized Empirical Likelihood Inferences For Some Semiparametric Regression Models With Censored Data

Regression analysis is an important area of statistical research. The paper studies several commonly used regression models when the response variable is randomly censoring of, by the means of the empirical likelihood method, and statistically inference the parameters in the regression model. Censored data is an important statistical data in research and reality of fields like medicine, reliability engineering, finance and insurance, environmental science and so on.For the regression model which the response variable is randomly censored, standard methods of regression analysis such as least squares method cannot be applied directly, so how to statistically analyse the regression model when there are censored data needs to be discussed in depth, and the study on the regression model when the response variable is randomly censored is of great significanceEmpirical likelihood method proposed by Owen (1988) is a non-parametric statistical method. Compared with traditional asymptotic normality method to construct confidence re-gion of parameters, the empirical likelihood method do not care about estimating asymptotic variance of parameters, which is an advantage of the empirical likelihood method. Further-more, the expression of asymptotic distribution variance of the parameter estimators in the model of the randomly censored regression is complex. Then, the application of empirical likelihood method is more meaningful.The paper studies the problem of parameter estimation in the regression model when the response variable is randomly censored, gives the empirical likelihood ratio statistic, and makes its asymptotic distribution is χ2distribution, avoids estimating asymptotic variance when constructing empirical likelihood confidence region of parameters, and improves the accuracy of the estimation.On the other hand, variable selection is one of the hot issues of the regression analysis research so far. Effective variable selection methods can select the remarkable variables and eliminate redundant variables to improve the prediction accuracy of the model. Tibshirani (1996) proposed LASSO penalty method, which is a coefficient shrunk method, and compared with the traditional subset selection method,the amount of its calculation is little and stable. At present, the coefficient shrunk method has been greatly concerned by statisticians, and some new penalty methods have been proposed to prove the Oracle property of selections.The paper studies variable selection and parameter estimation of the Cox proportional hazards model,which uses penalized empirical likelihood method combining coefficient shrunk method with empirical likelihood method.The main contents of this paper contain several following chapters.The second chapter investigates the question of the parameter estimation in non-linear semi-parametric regression model when the response variable is randomly right censored, constructs empirical log-likelihood ratio statistic and adjusted empirical log-likelihood ratio statistic for unknown parameters, proves that the constructed empirical likelihood ratio fol-lows an asymptotically χ2distribution under certain conditions, and constructs a confidence region of the unknown parameters. In addition, this chapter constructs least squares esti-mators of the unknown parameters, and proves its asymptotic properties. By corresponding simulation results, the empirical likelihood method is better than the least squares method at the coverage probability and accuracy of confidence region.The third chapter investigates the question of the parameter estimation in non-linear semi-parametric regression model when the response variable is randomly right censored and the nonparametric covariate has measurement error. An empirical log-likelihood ratio statistics for unknown parametric components is proposed, and it is proved that the pro-posed statistics follow an asymptotically χ2distribution under the null hypothesis, and the consequence can be used to construct the confidence region of the unknown parameter In addition, the least squares estimator of the unknown parameters is constructed, and its asymptotic properties is proved. Corresponding simulation results show that the empirical likelihood method is better than the least squares method at the coverage probability of the confidence region as well as precision.The fourth chapter mainly investigates the question of the estimation of the parame-ter part of semiparametric varying-coefficient partially linear errors-in-variables models in the condition of random right censored response variable. An empirical log-likelihood ratio statistics for unknown parametric components is proposed, and it is proved that the pro-posed statistics follow an asymptotically χ2under the null hypothesis, and the consequence can be used to construct the confidence region of the unknown parameter. By imitating, the confidence regions constructed by empirical likelihood method and the normal approxima-tion method are compared in terms of length of interval and coverage probability under the condition of finite sample. In the fifth chapter, the question of variable selected is researched with penalized em-pirical likelihood method in Cox proportional hazards model. The penalized function used is Bridge. The Oracle property of penalized empirical likelihood is discussed under certain conditions, namely, select the non-zero coefficients with probability1and the non-zero coef-ficients following a progressive normal distribution have the asymptotic normality. A penalty empirical likelihood ratio for regression coefficients is defined and it is proved to follow an asymptotically χ2distribution. Simulations and a real data example show that the proposed bridge penalty empirical likelihood have satisfying characters.The sixth chapter investigates the calculation of a kind of composite distribution in insurance and actuarial. The number of claim variable belongs to a widely distributed family, and claims amount follows a hybrid distribution. Firstly, present the recursive equation that composite distribution is satisfied. Secondly it is applied to the excess-of-loss reinsurance treaty to obtain corresponding recursive equation. Finally, give some concrete examples and numerical results.