Shape constrained estimation: Theory and methods

对函数的估计是当前许多统计方法的核心。这里有两种主要的竞争方法：参数估计和非参数估计。参数方法简单、高效、易于理解，但可能不是很健壮。非参数方法更稳健，因此更适用于中大样本量。基于核的估计器是一类流行的非参数方法，但这些方法需要用户选择带宽，这很难做到最优。另一种非参数方法，称为形状约束估计，是一种重要且极具吸引力的方法。 形状约束估计器(SCE)有两个关键属性：(1)它们是全自动的、自然的局部自适应的，并且不像其他基于非参数核的方法那样需要用户选择带宽；(2)形状约束通常是从手头的问题自然产生的(例如，累积分布函数必须是递增的，或者风险函数是先验已知递减的)。 形状约束方法是一种非常强大的统计工具。它允许人们指定一类分布，而不是单一的参数族，增加了结果的稳健性，往往以令人惊讶的适度效率损失为代价。此外，SCES的性能优于非参数核密度估计，并且已知是渐近最优的。SCES的应用包括检测混合的存在、基于模型的聚类、模式估计、回归方法、对竞争风险的推断以及ROC分析，仅举几例。 SCE通常是通过经验分布上的非线性算子得到的，因此计算和分析都很复杂。因此，目前对超临界流体力学的理论性质还缺乏完整的认识。拟议研究的预期结果包括在理论上对SCES的理解方面的进展，这将导致这一重要类别的稳健估计的新方法。

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.