Shape constrained estimation: Theory and methods

形状约束估计：理论与方法

• 批准号: 342858-2013
• 负责人: Jankowski, Hanna
• 金额: $ 1.09万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675649
• 关键词: Shape; constrained; estimation; Theory; methods

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.

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、有效、易于理解，但可能不可靠。 非参数方法更稳健，因此适用于中到大样本量。 基于核的估计是一类流行的非参数方法，但这些方法需要用户选择带宽，这很难做到最佳。 另一种非参数方法，称为形状约束估计，是一个重要的和非常有吸引力的选择。 形状约束估计器（SCE）有两个关键属性：（1）它们是全自动的，自然局部自适应的，并且不需要用户选择带宽，不像其他非参数基于核的方法，（2）形状约束通常从手头的问题中自然产生（示例包括必须增加的累积分布函数或先验已知为减少的风险函数）。 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，而不是一个单一的参数家庭，增加了结果的鲁棒性往往在一个令人惊讶的适度损失的效率。 此外，SCE可以优于非参数核密度估计，并且已知是渐近最优的。 SCE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、竞争风险的推断和ROC分析，仅举几例。 SCE通常通过经验分布上的非线性算子导出，因此计算和分析起来很复杂。 因此，一个完整的理解的理论性质的SCES是缺乏在这个时候。 拟议的研究的预期成果包括在理论上理解的SCE，这将导致新的方法，这一类重要的稳健估计的进展。