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

形状约束推理：不完整生存数据的测试和估计

• 批准号: RGPIN-2021-03124
• 负责人: Ling, HokKan
• 金额: $ 1.31万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759043
• 关键词: Shape; constrained; Inference; Testing; Estimation

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.

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。如果底层的参数形式是已知的，我们就有了最有效的估计方法，但由于可能存在模型规格错误，情况往往不是这样。鲁棒替代方案依赖于非参数方法。尽管核平滑方法已经得到了很好的研究，但由于形状约束下的估计通常是全自动的，不需要选择调优参数，因此近年来受到越来越多的关注。此外，由于科学知识和对基础物理理论的理论理解，这种定性约束往往是合理的。例如，在经济学中，效用和生产函数经常是递增和凹的。在密度估计中，对数凹密度类是单峰密度类的一个子集，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，根据定义，它们是非递减的。目前，在形状约束推理的文献中，有很大一部分集中在没有对是否存在这种形状结构进行诊断的估计上。此外，在实践中可能并不总是能够收集到感兴趣的基础变量的独立和同分布的数据。目前的研究计划旨在通过开发非参数似然、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。例如，这些涉及在非参数设置中构造似然比检验，以及扩展获得非参数最大似然估计的收敛速率的技术。该研究计划侧重于以下两个主题：(a)用于测试密度等函数的形状约束存在的通用方法；(b)不完全生存数据的非参数估计和半参数模型，如调查中经常收集的向后递归时间，其中从初始事件到调查抽样时间的时间被收集，但不涉及后续操作，因此所有失效时间都被删除，生存或风险函数的估计可以被视为形状约束估计问题。预期的研究成果将对形状约束推理的估计和检验的统计方法有重要的推动作用。为了使研究人员和实践者能够使用这些方法，将开发公开可用软件（如R）中的软件包。拟议的研究计划还将为高素质人才提供教育机会和支持。