Bootstrap and Threshold Models in Non-standard Problems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这项拟议的研究涉及在某些非参数场景中出现的一些非标准问题的方法和推理策略。“非标准”问题包括表现出非标准渐近性的情况--估计量收敛的速率不同于通常的平方根n速率和/或具有非正态极限分布。在这份提案中，研究者研究了统计研究的三个核心方向。这些是：（A）“非标准”问题中不同恢复方法的（In）-一致性，（B）对形状受限函数的估计和推断（其中在估计中结合了关于函数形状的知识，如单调性/凸性），以及（C）在发生急剧和潜在实质性变化（“状态变化”）的函数域中估计适当的“阈值”。在这些问题中存在固有的缺乏“平滑性”（有时称为“锐边效应”），表现在非标准收敛率和非正态极限分布中。在这些“非标准”问题中的统计推断是困难的，因为渐近分布理论是复杂的（在某些情况下是未知的），具有复杂的极限分布，包含讨厌的参数。Bootstrap方法是一种自然的选择，通常在“常规”平方根n收敛问题中是可靠的。虽然在过去的二、三十年里，人们已经广泛地了解了自举在不同“常规”场景中的行为，但在这种“非标准”问题上的工作并不多，这证明了在提案中进行的研究项目是合理的。天文学合作研究矮球状（dSph）星系中的暗物质含量和分布极大地刺激了对这些问题的研究。最近的估计表明，宇宙由大约96%的暗物质和暗能量组成，尽管人们对它们知之甚少。dSph星系在这项研究中占据了特殊的位置-它们被认为是包含暗物质的最小系统，因此对这些星系的研究在理解宇宙结构方面具有相当重要的意义。拟议中的研究还将有各种其他应用，从生物医学研究和流行病学等公共卫生学科到社会科学，特别是经济学。这是因为在分析企业/公司的产品时自然会出现“非标准”问题（经济学），研究随着年龄增长而死于疾病或感染的风险（生物医学研究），在婴儿早期发育（流行病学）的“敏感”时间段（影响健康）的调查，等等。水平课程。这种跨学科的研究将开辟在其他相关领域的普遍利益的调查途径。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This proposed research deals with methodological and inferential strategies in some non-standard problems that arise in certain non-parametric scenarios. The "non-standard" problems include situations exhibiting non-standard asymptotics -- where estimators converge at rates different from the usual square-root-n rate and/or have non-normal limit distributions. In this proposal, the investigator studies three core directions of statistical research. These are: (A) (In)-consistency of different resampling methods in "non-standard" problems, (B) Estimation and inference with shape restricted functions (where knowledge on the shape of the function, like monotonicity/convexity, is incorporated in estimation), and (C) Estimation of an appropriate "threshold" in the domain of a function where sharp and potentially substantial changes ("regime changes") occur. There is an inherent lack of "smoothness" in these problems (sometimes called the "sharp-edge effect") that manifests in the non-standard rates of convergence and the non-normal limit distributions. Statistical inference in these "non-standard" problems is difficult as the asymptotic distribution theory is complex (and in some cases unknown) with complicated limit distributions, containing nuisance parameters. Bootstrap methods are a natural alternative and are generally reliable in "regular" square-root-n convergence problems. Although there has been extensive activity in the last two/three decades in understanding the behavior of bootstrap in different "regular" scenarios, there has not been much work in such "non-standard" problems, justifying the research projects undertaken in the proposal. The study of the problems has been greatly stimulated by an astronomy collaboration investigating the dark matter content and distribution in dwarf spheroidal (dSph) galaxies. Recent estimates show that the universe consists of about 96% dark matter and dark energy, though very little is known about them as yet. The dSph galaxies occupy a special position in this study -- they are supposed to be the smallest systems containing dark matter, and hence the study of these galaxies is of considerable importance in understanding the structure of the universe. The proposed research will also have diverse other applications, ranging from disciplines in public health like biomedical studies and epidemiology to aspects of the social sciences, especially economics. This is because "non-standard" problems arise naturally in the analysis of productions of firms/companies (economics), the study of the risk of succumbing to illness or infection with age (biomedical research), in the investigation of "sensitive" time periods (affecting health) in the early development of infants (epidemiology), and so on. The frontiers of the proposed research can be extended through incorporation in the Ph.D. level curriculum. Such interdisciplinary research will open up avenues of investigation in other realted areas of universal interest.