Shape-constrained Inference: Testing and Estimation for Incomplete Survival Data

密度、分布函数和危险率等函数的估计一直是统计学中的一个基本问题。如果基本的参数形式是已知的，我们就有了最有效的估计方法，但情况往往并非如此，因为可能存在模型错误指定。稳健的替代方案依赖于非参数方法。虽然核平滑方法已经得到了很好的研究，但形状约束下的估计最近受到了更多的关注，因为它通常是全自动的，不需要选择调整参数。此外，由于科学知识和对基本物理理论的理论理解，这样的定性约束通常是合理的。例如，在经济学中，效用函数和生产函数往往是递增的和凹的。在密度估计中，对数凹密度类是单峰密度类的子集，它包含了大多数常用的参数分布，被认为是高斯密度类的自然无限维推广。其他例子包括估计分布函数或累积风险函数，其中它们的定义是非递减的。目前，在形状约束推理中，相当一部分文献集中在估计上，而没有对是否存在这样的形状结构进行任何诊断。此外，在实践中，可能并不总是可能收集到相关变量的独立和相同分布的数据。目前的研究方案旨在通过发展关于非参数似然、经验过程理论和半参数理论的新方法来解决形状约束推理中的这些挑战。这些包括，例如，在非参数设置中构建似然比检验，以及扩展获得非参数最大似然估计量的收敛速度的技术。该研究方案主要关注以下两个主题：(A)检验密度等函数是否存在形状约束的通用方法；(B)不完全生存数据的非参数估计和半参数模型，例如在调查中经常收集的向后复发时间，其中收集从初始事件到调查抽样时间的时间，但不涉及跟踪，使得所有的失败时间都被删除，生存函数或风险函数的估计可以归结为形状约束估计问题。预期的研究成果将大大促进形状约束推理中估计和检验的统计方法的发展。为了使研究人员和从业人员能够接触到这些方法，将开发R等公开可用的软件包。拟议的研究计划还将为高素质人员提供教育机会和支持。

Estimation of functions such as densities, distribution functions and hazard rates has always been a fundamental problem in statistics. If the underlying parametric form is known, we have the most efficient way of estimation, yet it is often not the case as there could be model misspecification. A robust alternative relies on nonparametric methods. Although kernel smoothing methods are well-studied, estimation under shape constraints has been receiving more attention recently because it is often fully automatic without a choice of tuning parameters. Moreover, such qualitative constraints are often plausible as a result of scientific knowledge and theoretical understanding of the underlying physical theory. For example, in economics, utility and production functions are often increasing and concave. In density estimation, the class of log-concave densities, which is a subset of the class of unimodal densities and contains most of the commonly used parametric distributions, has been regarded as a natural infinite-dimensional generalization of the class of Gaussian densities. Other examples include estimating a distribution function or a cumulative hazard function, where they are by definition nondecreasing. Currently, a significant portion of literature in shape-constrained inference focuses on estimation without any diagnosis on whether such shape structures are present. Besides, independent and identically distributed data of the underlying variable of interest may not always be possible to collect in practice. The current research proposal aims to address these challenges in shape-constrained inference by developing novel methodologies on nonparametric likelihood, empirical process theory and semiparametric theory. These involve, for example, constructing likelihood ratio tests in a nonparametric setting and extending techniques for obtaining rates of convergence of nonparametric maximum likelihood estimators. The research proposal focuses on the following two themes: (a) a universal approach for testing the presence of shape constraint for functions such as densities; and (b) nonparametric estimation and semiparametric models for incomplete survival data such as backward recurrence times often collected in surveys, in which time from an initial event to a survey sampling time is collected but no follow-up is involved so that all failure times are censored, and the estimation of survival or hazard functions can be cast as a shape-constrained estimation problem. The anticipated research outcomes will significantly advance the statistical approaches for both estimation and testing in shape-constrained inference. To allow the methodologies to be accessible to researchers and practitioners, packages in publicly available software such as R will be developed. The proposed research program will also provide educational opportunities and support for highly qualified personnel.