Bootstrap and Threshold Models in Non-standard Problems

这个奖项是根据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情景下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--估计器以不同于通常的平方根n率的速度收敛，和/或具有非正态极限分布。在这项提案中，研究者研究了统计研究的三个核心方向。它们是：(A)(In)--“非标准”问题中不同重采样方法的一致性；(B)形状受限函数的估计和推断(其中关于函数形状的知识，如单调性/凸性，被结合在估计中)；以及(C)在发生急剧和潜在的实质性变化(“制度变化”)的函数的域中估计适当的“阈值”。在这些问题中，存在着一种内在的“平稳性”(有时称为“尖端效应”)，表现为非标准的收敛速度和非正态的极限分布。这些“非标准”问题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方法是一种自然的替代方法，在“常规的”平方根-n收敛问题中通常是可靠的。尽管在过去的二三十年里，在理解不同“常规”情景下Bootstrap的行为方面进行了广泛的活动，但在这种“非标准”问题上没有做太多的工作，这证明了提案中所进行的研究项目是合理的。一个研究矮球星系中暗物质含量和分布的天文学合作极大地促进了对这些问题的研究。最近的估计表明，宇宙中约96%的暗物质和暗能量是由暗物质和暗能量组成的，尽管我们对它们知之甚少。DSph星系在这项研究中占有特殊的地位--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对于理解宇宙的结构具有相当重要的意义。拟议中的研究还将在其他方面得到广泛应用，从生物医学研究和流行病学等公共卫生学科，到社会科学的各个方面，特别是经济学。这是因为，在分析公司/公司的产品(经济学)、研究随年龄患病或感染的风险(生物医学研究)、调查婴儿早期发育的“敏感”时间段(影响健康)(流行病学)等方面，自然会出现“非标准”问题。拟议研究的前沿可以通过将其纳入博士水平的课程来扩展。这种跨学科的研究将在其他普遍感兴趣的相关领域开辟调查途径。

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.