排序集抽样下随机删失数据的非参数估计

Ranked set sampling protocol is appropriate for situations in which quantification of sampling units is difficult but ranking of the units is easy. There are large numbers of research achievements about nonparametric statistics of complete data under ranked set sampling. However, random censored data is often found in survival analysis. Therefore, it has important theoretical and practical significance to study nonparametric statistics of random censored data under ranked set sampling.. The project studies the nonparametric estimation of random censored data under ranked set sampling. The content of this project is divided into two parts. The first part is the product-limit estimation of the population survival function, which includes the proof of strong consistency and the comparison between estimation efficiencies. The second part is the kernel density estimation of the population probability density function, which includes the construction of estimator, the proof of strong consistency and asymptotic normality, and the comparison between estimation efficiencies. To analyze and infer the research content, the project need to use the properties of ranked set sample and the knowledge of probability and statistics, draw support from the previous research results, and adopt the means that combining theoretical analysis and numerical simulation.. The research results of this project not only offer the idea for the estimation of other survival index, such as hazard function, life expectancy and so on, but also lay the solid foundation for other statistical problems, such as hypothesis testing, regression analysis and so on. In addition, the research results have extensive application prospect in clinical medicine, ecological environment and so on.

排序集抽样方法适用于样本测量困难但排序容易的场合，该抽样下完全数据的非参数统计已有大量的研究成果。然而，在生存分析中经常碰到的是随机删失数据。因而，排序集抽样下随机删失数据的非参数统计研究具有重要的理论和实践意义。. 本项目研究排序集抽样下随机删失数据的非参数估计问题。研究内容分为两部分，一是总体生存函数的乘积限估计，包括强相合性的证明、估计效率的比较；二是总体概率密度函数的核密度估计，包括估计量的建立、强相合性和渐近正态性的证明、估计效率的比较。本项目利用排序集样本性质和概率统计知识，借助已有的研究结果，采用理论分析与数值模拟相结合的手段，对研究内容进行分析和推断。. 本项目的研究结果不仅为其它生存指标如风险率函数、平均寿命等估计问题提供了思路，也为其它统计问题如假设检验、回归分析等奠定了坚实的基础。另外，研究结果在临床医学、生态环境等领域都有广泛的应用前景。

排序集抽样是一种提高抽样效率的方法，适合样本易于排序但不易于测量的场合。总体生存函数和总体概率密度函数是生存分析中两个重要的度量指标，随机刪失数据是实际生活中经常出现的生存数据。为了提高估计效率，本项目利用排序集抽样下随机刪失数据，研究了总体生存函数和总体概率密度函数的非参数估计问题，研究结果如下：. (1) 为了估计未知总体的生存函数，提出了排序集抽样下随机删失数据的乘积限估计量，证明了新估计量的强相合性，确定了其强收敛速度，并与简单随机抽样下相应估计量进行估计效率的比较，结果表明排序集抽样效率高于简单随机抽样。最后，对生态环境的一组真实数据进行了实际应用，应用结果验证了排序集抽样方法的高效率性。. (2) 为了估计未知总体的概率密度函数，提出了基于随机刪失排序集样本的核密度估计量，证明了新估计量的渐近正态性和强相合性，确定了其渐近方差和强收敛速度，并与简单随机抽样下相应估计量进行了估计效率的模拟比较，结果表明排序集抽样效率高于简单随机抽样。. 本项目的研究结果为其它生存指标如风险率函数、平均寿命等估计问题提供了思路和基础。另外，研究结果在临床医学、环境科学、经济学等领域都有广泛的应用前景。

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