Bootstrap and Threshold Models in Non-standard Problems

非标准问题中的引导模型和阈值模型

• 批准号: 0906597
• 负责人: Bodhisattva Sen
• 金额: $ 10.01万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906597&HistoricalAwards=false
• 关键词: Bootstrap; Threshold; Models; Non; standard

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.

该奖项是根据2009年的《美国复苏和再投资法》（公法111-5）资助的。本提议的研究涉及在某些非参数场景中某些非标准问题中的方法论和推论策略。 “非标准”问题包括表现出非标准渐近性的情况 - 估计器以不同于常规的平方根率和/或具有非正常极限分布的速率收敛。在此提案中，研究人员研究了统计研究的三个核心方向。这些是：（a）（in） - “非标准”问题中不同的重采样方法的矛盾，（b）估计和推断形状受限函数的估计和推断（其中对功能形状的知识，例如单调性/凸性，在估计中都包含在估计中），以及（c）在官能变化中的适当“ threshold”的估计，在函数上发生了强大的更改和可能发生的变化。在这些问题（有时称为“锋利的效应”）中固有缺乏“平滑度”，这种问题表现出非标准的收敛速率和非正常极限分布。这些“非标准”问题的统计推断很困难，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，其中包含滋扰参数。引导方法是一种天然的替代方法，通常在“常规”平方根-N收敛问题中可靠。尽管在过去的两/三十年中，在了解不同的“常规”场景中的行为方面已经进行了广泛的活动，但在这种“非标准”问题中并没有太多工作，证明了该提案中进行的研究项目是合理的。天文学合作研究了矮球（DSPH）星系中的暗物质含量和分布，对问题的研究极大地刺激了。最近的估计表明，宇宙由大约96％的暗物质和深色能量组成，尽管对它们尚不清楚。 DSPH星系在这项研究中占据了一个特殊的位置 - 它们应该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非常重要。拟议的研究还将具有其他不同的应用，从公共卫生中的学科（如生物医学研究和流行病学）到社会科学的各个方面，尤其是经济学。这是因为“非标准”问题自然出现在对公司/公司（经济学）的分析，对疾病或年龄（生物医学研究）（生物医学研究）中屈服或感染的风险的研究，在研究婴儿早期发展（影响婴儿学）（流行病学）的“敏感”时期（影响健康）等。拟议的研究的前沿可以通过在博士学位中纳入。水平课程。这样的跨学科研究将在其他具有普遍关注的实用领域的调查途径开放。