题名 左截断右删失模型中的非参数统计推断
| 外文题名 | | Nonparametric statistical inference in the left truncated and right censored model |
该文主要是对左截断右删失模型进行非参数统计推断,建立了乘积限估计和累积失效率函数估计在一串只与次序统计量有关的递增区间上的i.i.d表示,其中余项的收敛速度只与此区间所包含的最大次序统计量的下标有关,并由此推出了乘积限估计和累积失效率函数估计的一些渐近性质.对随机右删失数据下的四种不同的失效率函数估计进行研究,在较弱条件下,改进并扩充了现有文献的结果.利用乘积限估计的鞅表示分别获得了概率密度估计和分位点密度估计的极大偏差的渐近分布.在此基础上,根据停时规则,分别建立了概率密度和分位点密度的序贯区间估计,并证明了当区间长度趋于零时,置信区间的置信水平可任意接近于1.建立了分位点函数及其导数的核估计的Bahadur类强表示,获得了余项的精确收敛速度并由此导出了这些估计的重对数律和渐近正态性.获得了乘积限过程和累积失效率过程的振动模和Lipschitz-1/2模的强一致收敛的精确速度以及这两个过程的局部振动模和局部Lipschitz-1/2模的逐点收敛的精确速度.作为定理的应用,推导了各种核密度估计和失效率估计的强一致收敛和逐点收敛的精确速度.
姓名 孙六全
院系 概率系
专业 概率论与数理统计
第一导师姓名 郑忠国
第一导师单位 北京大学 概率系
关键词 非参数统计 数理统计 左截断右删失 乘积限估计
分类号 BSLW /1999 /O212.7 /6
摘要
本文主要是对左截断右删失模型进行非参数统计推断，全文共分五章
第一章第1.1节建立了乘积限估计和累积失效率函数估计在一串只与次序统计量有关的递增区间上的i.i.d表示，其中余项的收敛速度只与此区间所包含的最大次序统计量的下标有关，并由此推出了乘积限估计和累积失效率函数估计的一些渐近性质。
第1.2节给出了一类重要的广义Von-Mises泛函估计的强逼近和U-统计量表示，并由此得到了此类泛函估计的强相合性，渐近正态性以及重对数律
第二章第2.1节对随机右删失数据下的四种不同的失效率函数估计进行研究，在较弱条件下，改进并扩充了现有文献的结果。获得了这四种估计的渐近正态性，一致强弱相合收敛速度以及重对数律，并通过数值模拟来说明这四种估计没有显著性的差异。
第2.2节建立了失效率函数核估计的平方积方误差的中心极限定理和均方积分误差的渐近展开。
第2.3节借助于失效率函数核估计的平方积方误差的渐近表示和鞅不等式，通过最小平方交叉核实的方法来进行窗宽选择，并证明了由此选择的窗宽在某种意义下是渐近最优的。
第三章第3.1节和第3.2节利用乘积限估计的鞅表示分别获得了概率密度估计和分位点密度估计的极大偏差的渐近分布。在此基础上，根据停时规则，分别建立了概率密度和分位点密度的序贯区间估计，并证明了当区间长度趋于零时，置信区间的置信水平可任意接近于1。
第四章第4.1节建立了分位点函数及其导数的核估计的Bahadur类强表示，获得了余项的精确收敛速度并由此导出了这些估计的重对数律和渐近正态性。第4.2节在随机右删失数据下讨论了由核光滑的乘积限过程和它的分位点过程所构成的Bahadur-Kiefer过程的渐近分布。证明了在不同的窗宽选择下，所得到的渐近分布可以完全不同，并由此得出了它的所有可能的渐近分布。 第五章第5.1节获得了乘积限过程和累积失效率过程的振动模和Lipschitz-〓模的强一致收敛的精确速度以及这两个过程的局部振动模和局部Lipschitz-〓模的逐点收敛的精确速度。作为定理的应用，推导了各种核密度估计和失效率估计的强一致收敛和逐点收敛的精确速度。第5.2节在较一般情况下建立了累积失效率过程的局部振动模的泛函重对数律，并由此导出了失效率核估计的重对数律。
外文摘要 THE LEFT TRUNCATED AND RIGHT CENSORED MODEL Abstract
In this dissertation,we mainly deal with nonparametric statistical inference inthe left truncated and right censored model.<br/> In Setion 1.1 of Chapter 1,strong representations of the cumulative hazardfunction estimator and the product-limit(PL)estimator of the survival function arederived,which are valid up to a given order statistic of the observation.A precisebound for the errors is obtained which only depends on the index of the last orderstatistic to be included.In Section 1.2 of Chapter 1,the estimation of a generalclass of Von-Mises type functionals of the survival function is investigated.A strongapproximation and an almost sure asymptotic representation of the estimator areestablished.In addition,the asymptotic normality,the strong consistency and thelaw of iterated logarithm for the estimator are obtained.<br/> In Section 2.1 of Chapter 2,several estimators of hazard rate function are pro-posed by different ways when the data are subject to right censoring.The asymptoticnormality,the strong and weak consistency and the laws of iterated logarithm forthese estimators are derived under some weak conditions.The limited simulationresults show that these estimators have no significant difference.In Section 2.2of Chapter 2,a central limit theorem for the integrated square error of the kernelhazard rate estimators is obtained,and an asymptotic representation of the meanintegrated square error for the kernel hazard rate estimators is also presented.InSection 2.3 of Chapter 2,based on an asymptotic representation of the integratedsquare error for the kernel hazard rate estimators and a martingale inequality,itis shown that the bandwidth selected by the data-based method of least squarecross-validation is asymptotically optimal in a compelling sense.<br/> In Sections 3.1 and 3.2 of Chapter 3,based on the martingale integral represen-tation of the product-limit estimator,asymptotic distributions are obtained for themaximal deviation of the kernel density estimator from the density and for maximaldeviation of the kernel quantile density estimator from the quantile density.Usingthese results and the stopping rules,we propose a fully sequential procedure for con-structing fixed-width confidence bands for the density and the quantile density ona finite interval,and show that the procedure has the desired coverage probabilityasymptotically as the width of band approaches zero.<br/> In Section 4.1 of Chapter 4,the Bahadur type representations are establishedfor the kernel quantile estimators and the kernel estimators of the derivatives of thequantile function.Under suitable conditions,with probability one,the exact conver-gent rates of the remainder terms in these representations are obtained.From theserepresentations,the asymptotic normality and the laws of the iterated logarithmfor these kernel estimators are derived.In Section 4.2 of Chapter 4,the asymp-totics of Bahadur-Kiefer process based on smoothed PL and its quantile processesare considered when the data subject to right censoring.It is shown that with thedifferent chioce of smoothing parameter,the asymptotic distribution on smoothedBahadur-Kiefer process can be completely different.A complete characterization ofthe possible limits can be obtained.<br/> In Section 5.1 of Chapter 5,the exact rates of uniform convergence are ob-tained for oscillation moduli and Lipschitz -〓 moduli of PL process and cumulativehazard process,and the exact rates of pointwise almost sure convergence are alsoestablished for local oscillation moduli and local Lipschitz-〓 moduli of PL processand cumulative hazard process.Based on these results,the exact rates of uniformconvergence and pointwise almost sure convergence for various types of density andhazard function estimators are derived.In Section 5.2 of Chapter 5,functional lawsof the iterated logarithm are obtained for cumulative hazard process in the neighbor-hood of a fixed point.Based on these results,the pointwise strong limiting behaviorof the kernel hazard rate estimator is derived.<br/>
研究领域 数理统计及其应用
总页码 159
参考文献总数 0
答辩日期 1998-1-1
入学年份 1900
馆藏 002/98-<01>
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