Mathematical Sciences: Leveraged Bootstrap

DMS 96-26532 REN本研究的目的是研究一种新的重抽样方法，称为杠杆自举。杠杆Bootstrap便于使用各种类型的删失数据，包括右删失数据、双重删失数据、区间删失数据等，来研究广泛的非参数和半参数统计量。该方法不依赖于将通常的完全数据的统计量扩展到不完全数据，具有简单、计算效率高、易于应用于具有不同类型删失数据的统计推断问题。在本研究中，我们考虑了以下几个问题：(1)杠杆自举在实践中的应用；(2)杠杆自举的效率；(3)与其他方法的比较。具体地说，研究了杠杆Bootstrap在一些统计推断问题中的应用，如构造置信带和检验的经验似然方法，以及与几个统计模型相关的假设检验问题。在统计文献中，各种类型的删失数据通常被称为不完整数据。例如，死于心脏病的患者不能继续死于肺癌。在这种情况下，患者的肺癌生存时间是不完整的。最近，在一些非常重要的临床试验中，如乳腺癌研究和艾滋病研究，都遇到了双删失数据和区间删失数据。对这类删失数据的统计研究总体上仍然落后于对删失数据的统计研究。使用这些复杂类型的删失数据进行统计分析的主要困难在于，为完整数据开发的常用方法通常不能直接扩展到不完整数据，并且在某些情况下，通常的非参数自举方法可能会非常耗时。本研究的目的是开发一些新的统计方法，这些方法准确、计算高效、易于适用于不同类型的删失数据。***

DMS 96-26532 Ren The objective of this research is to investigate a new resampling method, called the Leveraged Bootstrap. The Leveraged Bootstrap facilitates research in a broad class of nonparametric and semiparametric statistics using various types of censored data, including right censored data, doubly censored data, interval censored data, etc.. This method does not depend on the extensions of the usual statistics for complete data to incomplete data, and is simple, computationally efficient and easily applicable to a wide of variety of statistical inference problems with different types of censored data. In this research, the following issues are considered: (i) how the leveraged bootstrap should be applied in practice; (ii) the efficiency of the leveraged bootstrap; (iii) comparison with other methods. Specifically, the investigator studies the application of the leveraged bootstrap in some statistical inference problems such as the empirical likelihood methods in constructing confidence bands and tests, and the hypothesis testing problems which are associated with several statistical models. %%% Various types of censored data are generally referred to as incomplete data in statistics literature. For instance, a patient who has died from heart disease cannot go on to die from lung cancer. In such a case, the survival time of lung cancer of the patient is incomplete. Recently, doubly censored data and interval censored data have been encountered in some very important clinical trials such as breast cancer research and AIDS research. The statistical research on these types of censored data still generally lags behind that on right censored data. The principle difficulties in statistical analysis using these complicated types of censored data are that the usual methods developed for complete data often do not have direct extensions to incomplete data, and that the usual nonparametric bootstrap can be quite computationally time consuming in certain cases. This research is to develop some new statistical methods which are accurate, computationally efficient and easily applicable for different types of censored data. ***