Empirical Likelihood Inferences On Semiparametric Models

This dissertation studies statistical inferences on semiparametric models based on either negatively associated random errors or censored data based on empirical likelihood method. The main contents include the following three as-pects.Firstly, we obtained empirical likelihood ratio statistics based on estimating equations for partial linear models under negatively associated random errors. Using the blockwise empirical likelihood with large block and small block, we proved the statistics is asymptotically chi-squared distributed with freedom equal to the dimension of the parameter of interest. We can build the confidence region for the parameter using this theorem, or making hypothesis tests on the parameter. Finally, we conducted a small sample simulation. The simulation results showed that the blockwise empirical likelihood are more attractable than the ordinary empirical likelihood in terms of higher coverage probability of the confidence region.Secondly, we made large sample statistical inferences on single-index re-gression models with negatively associated random errors. The score function we use is bias corrected. It is because of the bias correction, that the empirical likelihood ratio statistics we constructed converges to a standard chi-squared dis-tributed random variable, instead of the weighted sum of chi-squared distribution form due to the score function without bias correction. By our conclusion, the estimation of the confidence region of the index in the single-index model can be conducted.Thirdly, we deal with the censored data often appeared in the survival anal-ysis. We made statistical inferences on the partial linear model under random right censoring. We constructed the score function based on the Buckley-James estimator. Using the martingale representation of the score function, we can rewrite the score function as a partial sum of the counting process martingale by means of the relationship between the counting process martingale and the Doob’s expectation martingale. Utilizing the common theory of the linear rank statistics and the Rebolledo martingale central limit theorem, we proved the em-pirical likelihood ratio statistics has asymptotic chi-squared distribution. This conclusion can be used to construct the confidence region of the parameter of in-terest. We also conducted a simulation which illustrated that the Buckley-James estimator is superior to the KSV estimator.