不完全信息比例风险模型极大似然估计的极限理论
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摘要
本文研究不完全信息随机右删失场合比例风险模型未知参数的极大似然估计的大样本性质.比例风险模型是生存分析里重要的回归模型之一,不完全信息随机右删失数据是生存分析里常见的数据类型,因此研究不完全信息随机右删失场合比例风险模型是十分有意义的.目前尚未见文献涉及不完全信息随机右删失数据下比例风险模型的研究,该模型的极大似然估计的理论性质更属未知.本文在适当的假设下,得到了不完全信息随机右删失场合比例风险模型未知参数极大似然估计的相合性、渐近正态性、重对数律及中偏差等大样本性质,大部分结果即使是在传统的比例风险模型中也是首次得到.
本文由五章构成：
在第一章中.我们先简要介绍研究背景,前人已有的工作,本文的主要工作及意义.然后建立了不完全信息随机右删失比例风险模型,给出了样本的似然函数.
在第二章中,我们在协变量非随机的假定下,给出了不完全信息比例风险模型未知参数β的极大似然估计的存在性、相合性和渐近正态性.
在第三章中,我们在前一章的基础上得到了极大似然估计满足的重对数律、Chung重对数律和中偏差.
在第四章中,在协变量为随机变量的情形下,不完全信息随机右删失场合比例风险模型的极大似然估计的存在性、相合性和渐近正态性、重对数律及中偏差等结果被一一导出.
第五章,讨论了不完全信息随机右删失场合数量性状位点(QTL)的定位问题,提出了相应的区间定位方法,并通过数值模拟验证了方法的有效性.
We address some important topics in the large-sample theory for the proportional hazards model based on the data with random censorship and incomplete information. In survival analysis the proportional hazards model is a significant regression model and right censoring data with incomplete information is a common data type. The research on proportional hazards model randomly censored with incomplete information is very meaningful. Under suitable assumptions, we obtain the consistency, asymptotic normal-ity, law of iterated logarithm and the moderate deviation of the maximum likelihood estimator of unknown parameters. Most of the conclusions are the first time to be ob-tained.
This thesis consists of five parts as follows:
In chapter one, We first provide a brief description of the background, previous work, the main results of our work. Then, we introduce the proportional hazards model randomly censored with incomplete information and give the likelihood function.
In the second chapter, we obtain the existence, consistency and asymptotic nor-mality of the maximum likelihood estimator under the hypothesis that covariates are non-random.
In the third chapter, we prove that the maximum likelihood estimator which ob-tained in the chapter two satisfy the law of iterated logarithm, Chung type law of iterated logarithm and the moderate deviation.
In chapter four, under the presumption that covariates are random, the results of the existence, consistency and asymptotic normality, the law of iterated logarithm deviation and the moderate deviation of the maximum likelihood estimator are obtained one by one.
In the final chapter, we discuss how to map the quantitative trait loci based on the data randomly censored with incomplete information and propose a corresponding interval mapping method which verified by numerical simulation method.
本文由五章构成：
在第一章中.我们先简要介绍研究背景,前人已有的工作,本文的主要工作及意义.然后建立了不完全信息随机右删失比例风险模型,给出了样本的似然函数.
在第二章中,我们在协变量非随机的假定下,给出了不完全信息比例风险模型未知参数β的极大似然估计的存在性、相合性和渐近正态性.
在第三章中,我们在前一章的基础上得到了极大似然估计满足的重对数律、Chung重对数律和中偏差.
在第四章中,在协变量为随机变量的情形下,不完全信息随机右删失场合比例风险模型的极大似然估计的存在性、相合性和渐近正态性、重对数律及中偏差等结果被一一导出.
第五章,讨论了不完全信息随机右删失场合数量性状位点(QTL)的定位问题,提出了相应的区间定位方法,并通过数值模拟验证了方法的有效性.
We address some important topics in the large-sample theory for the proportional hazards model based on the data with random censorship and incomplete information. In survival analysis the proportional hazards model is a significant regression model and right censoring data with incomplete information is a common data type. The research on proportional hazards model randomly censored with incomplete information is very meaningful. Under suitable assumptions, we obtain the consistency, asymptotic normal-ity, law of iterated logarithm and the moderate deviation of the maximum likelihood estimator of unknown parameters. Most of the conclusions are the first time to be ob-tained.
This thesis consists of five parts as follows:
In chapter one, We first provide a brief description of the background, previous work, the main results of our work. Then, we introduce the proportional hazards model randomly censored with incomplete information and give the likelihood function.
In the second chapter, we obtain the existence, consistency and asymptotic nor-mality of the maximum likelihood estimator under the hypothesis that covariates are non-random.
In the third chapter, we prove that the maximum likelihood estimator which ob-tained in the chapter two satisfy the law of iterated logarithm, Chung type law of iterated logarithm and the moderate deviation.
In chapter four, under the presumption that covariates are random, the results of the existence, consistency and asymptotic normality, the law of iterated logarithm deviation and the moderate deviation of the maximum likelihood estimator are obtained one by one.
In the final chapter, we discuss how to map the quantitative trait loci based on the data randomly censored with incomplete information and propose a corresponding interval mapping method which verified by numerical simulation method.
引文
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[11]陈怡南,叶尔骅.带有不完全信息随机截尾试验下Weibull分布参数的MLE.数理统计与应用概率,1996,11(4)：353-363.
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[14]陈家鼎.生存分析与可靠性.2005,北京,北京大学出版社.
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[30]樊军、高付清.线性模型的最小二乘估计的中偏差及其重对数律.数学杂志,2007,1,60-64.
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[35]Gao, F Q. Moderate Deviations for the Maximum Likelihood Estimator. Statistics and prob-ability Letters,2001,55,345-352.
[36]Gao, F Q. Moderate Devitaions and Large Deviations for Kernel Density Estimators. Journal of Theoretical Probability,2003,16,401-418.
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