Monotone empirical bayes, non/semiparametric methods for nonlinear/hazards/censored regression and functional estimation

在许多领域，如医学、农业、工业、工程、生物科学、社会学，经常会出现涉及相似但独立的调查序列的情况。在这种情况下，当序列以未知的概率分布进行时，感兴趣的参数经常不可预测地变化，因此不能做出通常所称的贝叶斯决策的最小风险决策。然而，从以前的调查中收集的信息有时可以用来制定决策，即通常所说的经验贝叶斯决策，其风险接近最小贝叶斯风险。这项研究的部分目的是提出改进的/单调EB估计/检验程序，当响应由某些参数分布建模时，并调查这些程序的风险接近最小贝叶斯风险的速度和最佳可能速度。当响应是患者的寿命、失败时间等事件的时间结果时，数据通常被删失，并且涉及时间依赖协变量。例如，在一项临床试验中，研究人员感兴趣的是，根据BMI(身体混乱指数)和年龄等其他预测协变量进行调整后，评估治疗对HIV-1血清阳性吸毒者生存的影响。当研究结束时，一些患者可能还活着。因此，这些患者的生存时间是被删失的。本研究的目的之一是在样本被删失的情况下为协变量效应建立改进的非参数/半参数回归模型，并研究其效率和最优收敛速度。在几乎每个学科的响应数据依赖于几个因果协变量，人们经常面临为预测/预测目的而对这些协变量上的响应数据进行建模的问题。这项研究建议在一定程度上是为了扩展预测/预测的方法，而不需要对响应对协变量的依赖(回归)的函数形式进行任何说明，其中这些变量可以像经济学中那样随机地独立(或时间)依赖，也可以像生存分析中那样是终身数据。

In many fields, like medicine, agriculture, industry, engineering, biological sciences, sociology, often situations involving sequences of similar but independent investigations arise. In such situations the parameter of interest often varies unpredictably as the sequence progresses with unknown probability distribution, and hence a minimum risk decision, what is usually called Baysian decision, can not be made. However, the information collected from the previous investigations can sometimes be utilized to formulate a decision, what is popularly known as empirical Bayes decision, with risk close to the minimum Bayes risk. Part of the objective of this research is to propose improved/monotone EB estimation/test procedures when the responses are modelled by some parametric distribution and to investigate the speed and the best possible speed with which the risk of these procedures approach to the minimum Bayes risk.  When the response is the time-to-event outcome such as lifetimes of patients, failure times, the data are often censored and involve time dependent covariates. For example, in a clinical trial, the researchers are interested in evaluating the effect of a treatment on survival in the HIV-1 seropositive drug users adjusted for other predictive covariates such as BMI (body mess index) and age. Some patients may still be alive when the study terminates. Hence, the survival time of these patients are censored.  Part of the objective of this research proposal is to develop improved non/semi-parametric regression models for covariate effects with censored samples and to investigate the efficiency and the optimal convergence rates.In almost every discipline response data depend on several causal co-variates, and one is often faced with the problem of modelling the response data on these co-variates for forecasting/prediction purpose. This research proposal is in part to extend the methods of forecasting/prediction  without any specification of the functional form of the dependence (regression) of the responses on covariates, where these variables may be stochastically  (or time) dependent as in economics or may be life time data as in survival analysis.