Mathematical Sciences: Polynomial Spline Modeling in Survival Analysis and Stationary Stochastic Processes

该提案有四个项目，涉及应用对数样条方法的生存分析和平稳随机过程。第一个项目的主要目标是系统地扩展生存分析的各个方面，以处理涉及删失响应和协变量的多变量数据，并以有效平衡模型偏倚和估计方差的方式进行。第二个项目考虑了双变量截尾失效时间下的生存和风险函数的估计问题。由于这是生存分析中一个新的重要研究领域，该项目进一步分为两个子项目。第一个考虑没有协变量的二元生存函数的估计，第二个考虑涉及协变量的回归分析。我们的目标是为双变量生存分析提供一个一致的和统一的非参数框架。特别是，所提出的程序的渐近性质将被建立。第三个项目考虑了平稳过程的可能混合谱分布的估计。目的是建立连续谱和离散谱的联合估计的渐近性质。扩展到其他平稳过程，如空间和点过程也将得到强调。第四个项目考虑非参数时间序列回归。它的目的是研究回归函数估计的渐近性质，具有指定的最多d个变量的函数和的形式。本提案中的第一个项目涉及开发几种统计程序，用于识别和评估与冠心病、乳腺癌、肺癌、前列腺癌和结肠癌等各种疾病的发生和复发相关的重要风险因素。第二个项目研究这些风险因素对几种疾病的病因之间关系的影响。具体而言，该方法将应用于研究（1）糖尿病患者的激光光凝治疗，（2）非致命性心肌梗死和心力衰竭的生存概率，（3）结肠癌和乳腺癌患者的复发与生存时间的关系。第三个项目开发用于研究神经系统元素的统计程序，并了解它们如何发挥作用和协同工作。通过这些发展，我们希望解释记忆、情感、学习、睡眠和期望等问题。第四个项目开发了有效的预测程序，用于环境研究，如温度对河流流量的影响，臭氧浓度。

The proposal has four projects that involve the application of logspline methodology to survival analysis and stationary stochastic processes. The primary objective of the first project is to systematically extend survival analysis in all of its various aspects to handle multivariate data involving censored responses and covariates and to do so in a manner that will efficiently balance model bias and variance in the estimates. The second project considers the problem of estimating survival and hazard functions for bivariate censored failure times. Since this is a new and important research area in survival analysis, the project is further divided into two subprojects. The first considers the estimation of the bivariate survival function without covariates, and the second considers regression analysis involving covariates. The goal is to provide a consistent and unified nonparametric framework for bivariate survival analysis. In particular, asymptotic properties of the proposed procedures will be established. The third project considers the estimation of a possibly mixed spectral distribution of a stationary process. The objective is to establish asymptotic properties for the joint estimator of continuous and discrete spectra. Extensions to other stationary processes such as spatial and point processes will also be emphasized. The fourth project considers nonparametric time series regression. Its objective is to study the asymptotic properties of regression function estimates subject to having the form of a specified sum of functions of at most d variables. The first project in this proposal involves the development of several statistical procedures for identifying and evaluating important risk factors that are related to the occurrence and recurrence of various diseases such as coronary heart disease, breast, lung, prostate and colon cancers. The second project studies the effects of these risk factors on the relationship between causes of several diseases. Specifically, the method will be appl ied to study (1) the laser photocoagulation eye treatment for diabetic patients, (2) survival probabilities of nonfatal myocardial infarction and heart failures, (3) the relationship of recurrence and survival times of colon or breast cancer patients. The third project develops statistical procedures for studying elements of the nervous system and to understand how they function and work together. Through this developments, we hope to explain things such as memory, emotion, learning, sleep and expectation. The fourth project develops efficient prediction procedures for applications in environmental studies such as temperature effect on riverflow, ozone concentration.