Nonparametric Estimation and Inference: Shape Constraints, Model Selection, and Level Set Estimation

Larger and more complex data sets are becoming more and more commonplace. It is thus both advantageous and necessary to use statistical methods that are very flexible, and which allow the data to 'speak for itself,' rather than having researchers make strong unjustifiable prior assumptions about the data. Flexible methods are thus necessary in the modern landscape, but they have the difficulty that the practitioner generally must 'tune' the methods in order to get reliable results. This introduces an ad-hoc element to data analysis, leads to a lack of replicability in results, and means (incorrectly tuned) statistical procedures may return incorrect results. The unifying theme of this project is the development of statistical methods that are both very flexible and also fully automated, meaning they do not depending on user-chosen tuning parameters. The application areas motivating this project are varied, and include the analysis of vaccine trials, the study of economic data, and the problem of outlier detection (used widely in financial services).

Very flexible nonparametric statistical methods have become necessary tools to handle the complex nature of large data sets. One difficulty with using nonparametric tools in practice is their general dependence on (potentially many) tuning parameters which must be chosen well to ensure reliable performance. The focus of this proposal is on developing methods which can be implemented and lead to reliable results without requiring any ad-hoc steps by the end user. Three main problems will be studied: (a) model selection for shape-constrained estimators, (b) likelihood ratio type tests for shape-constrained estimators, and (c) estimation and inference for complex features of multivariate densities. In (a) and (b) the focus is on using so-called shape-constrained estimators, which have the benefit of simultaneously being nonparametric but also of automatically selecting optimal tuning parameters. Furthermore, they arise out of natural or axiomatic prior information (e.g., economic theory or laws of physics) in many settings, and in such cases one should certainly use that information. In (c), the focus is on estimation of complex features of densities (such as level set manifolds, motivated by outlier detection problems). Both shape-constrained methods and alternative methods will be considered, when shape constraints are not applicable. There are few or no effective procedures available in many of the problems under consideration, because of the nonstandard nature of the problems.