Shape constrained estimation: Theory and methods

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Estimation of a function lies at the core of much of current statistical methodology. There are two main competing methods here: parametric and nonparametric estimation. Parametric methodology is simple, efficient, well-understood, but may not be robust. Nonparametric methods are more robust, and therefore preferable for medium to large sample sizes. Kernel-based estimators are one popular class of nonparametric methods, but these require the user to select a bandwidth, which is difficult to do optimally. An alternative nonparametric approach, called shape-constrained estimation, is an important and highly attractive option. Shape-constrained estimators (SCEs) have two key attributes: (1) they are fully automatic, naturally locally adaptive, and do not require the user to select a bandwidth unlike other nonparametric kernel-based methods, and (2) shape constraints often arise naturally from the problem at hand (examples include cumulative distribution functions which must be increasing or hazard functions which are a priori known to be decreasing). Shape-constrained methodology is a very powerful statistical tool. It allows one to specify a class of distributions instead of a single parametric family, increasing the robustness of the results often at a surprisingly modest loss of efficiency. Furthermore, SCEs can outperform the nonparametric kernel density estimate and are known to be asymptotically optimal. Applications of SCEs include detecting the presence of mixing, model-based clustering, mode estimation, regression methods, inference for competing risks, and ROC analysis, to name only a few. SCEs are most often derived via nonlinear operators on the empirical distribution, and are hence complex to compute and analyze. Therefore, a complete understanding of the theoretical properties of SCEs is lacking at this time. The expected outcome of the proposed research includes advances in theoretical understanding of SCEs which will lead to new methodologies for this important class of robust estimators.