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

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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.