Mathematical Sciences: Weighted Empiricals in Regression with Survival Data

摘要：本研究涉及多种半参数回归模型，采用右删减、双删减或区间删减。这些模型包括常用的乘法风险模型和加速寿命的对数变换模型，较新的脆弱性模型，以及研究较少的乘法比值比模型。在所有这些模型中，允许协变量与时间相关。通过加权经验过程和一些自洽方程，定义了加权Nelson-Aalen风险函数和Kaplan-Meier生存函数。利用这些加权函数建立了各种估计方程和估计函数。然后得到并分析了回归估计的一些类别，包括秩估计、最小距离估计和m估计。研究了估计器的有效性和鲁棒性，模型的拟合优度检验，以及计算这些估计器的实际问题。本研究所研究的模型和相关统计问题在医学随访研究和工业可靠性检验等领域具有应用价值。在涉及这些领域的情况下，由于各种审查，可用的数据往往是不完整的。审查归因于患者或机器操作系统的条件，并受到诸如治疗，年龄，压力和负荷等因素的影响。这些影响可以通过本研究中的一些模型来表述。通过对这些模型和相关统计问题的研究，可以更好地理解、预测和控制患者或操作系统的状况。与现有的研究文献相比，本研究中的新方法将需要不同或更复杂的计算算法。这些方法的实现为高效编码和高性能计算提供了背景，并为其提供了便利。

Proposal: DMS 9504918 PI: Song Yang Institution: Texas A&M Title: Weighted Empiricals in Regression with Survival Data Abstract: This research involves various semiparametric regression models, with right, double or interval censoring. These models include the commonly used multiplicative hazard model and the log transform of accelerated life model, the relatively new frailty model, and the rarely studied multiplicative odds ratio model. In all these models, the covariates are allowed to be time-dependent. The weighted Nelson-Aalen hazard function and Kaplan-Meier survival function are defined via weighted empirical processes and some self consistency equations. Various estimating equations and estimating functions are established using these weighted functions. Some classes of the regression estimators, including the rank, minimum distance and M-estimators, are then obtained and analyzed. The efficiency and robustness of the estimators and some goodness-of-fit tests of the models is studies, as well as the practical issues on computing these estimators. The models and related statistical problems studied in this research have applications in areas such as medical follow-up studies and industrial reliability tests. In situations involved in those areas, the available data are often incomplete due to various censoring. The censoring is attributed to the conditions of the patients or machine operating systems and is influenced by factors such as treatment, age, stress, and load. These influences can be formulated by some of the models in this research. Through studies of these models and related statistical problems, the conditions of the patients or operating systems can be better understood, predicted, and controlled. Compared with the existing research literature, the new methods in this research will require different or more sophisticated computing algorithms. Implementations of these methods provide a background to, and are facilitated by, efficient coding and high performance computing.