Function estimation under shape constraints and detection of thresholds in nonparametric and semiparametric problems

该项目涉及非参数和半参数问题的方法和推理策略，这些问题表现出非标准渐近性，即估计器以不同于通常速率(样本大小的平方根)和/或具有非高斯极限分布的速度收敛的问题。研究的两个核心领域是：(A)估计形状限制下的函数；(B)估计发生急剧和潜在重大变化的函数域中的适当“阈值”。形状-限制性推理的部分重点将放在基于最大似然/最小二乘的程序上，特别是为感兴趣的数量(如回归函数、风险函数等)构建置信集。通过残差平方和或似然比统计量的倒置。这是考虑到对单调函数模型的初步探索，其中这种统计被认为表现出渐近枢轴行为，这有助于推断，因为不需要从数据估计滋扰参数。感兴趣的形状约束是单调性、单峰性和凸凹性，并且通常是这些特征的组合。形状约束下的平滑也将在这些问题中的一些中被研究，因为平滑通常产生更快的收敛速度，并提供函数的导数的自动估计，例如，在经济学中经常感兴趣的函数的导数。在门槛估计方面，将探索两个主要领域。第一种是变化点估计，其中感兴趣函数的值或导数的值出现跳跃，重点放在设计问题上：如何设计采样机制，在给定固定预算(可采样点)的情况下，以便精确检测跳跃。第二个领域涉及研究光滑函数的阈值的适当概念，这些函数在较短的区域内表现出快速变化。一种这样的概念可以用具有有限数量的不连续的函数来表述，该函数只是用作近似，或者是真实函数的工作模型。最佳拟合工作模型的不连续性提供了阈值的自然描述。根据一些最初的工作，裂点估计被证明与变点估计完全不同，将进行更详细的研究。拟议的研究将有不同的应用，从公共卫生的学科，如生物医学和流行病学，到社会科学(经济学)和物理科学(天文学)的主题。例如，形状限制自然会出现在公司/公司的产品分析(经济学)、疾病或感染随年龄增长的风险研究(生物医学研究/公共卫生)，以及与探测星系中暗物质相关的问题中，这些问题的解决方案可以揭示宇宙未来的演化(天体物理学)。P.I.积极参与与计量经济学家、流行病学家和天文学家的合作：因此，这项拨款产生的研究将具有强烈的跨学科色彩，并将解决许多真正令人感兴趣的科学问题。在阈值估计方面，分割点检测的统计方法很重要，因为生态学家在制定污染控制标准时已将分割点用作阈值的衡量标准。将与变点检测相关的问题应用于工程系统中的应力阈值检测。将制定的统计方法将通过免费软件包分发给学术界和工业界的统计界。在教育方面，这项建议中的一些研究材料将纳入研究生一级的高级课程和跨学科研讨会系列。一些项目还将作为博士研究生的论文主题，因此将在未来统计学家的培训中发挥重要作用。

The proposed project deals with methodology and inference strategies for nonparametric and semiparametric problems exhibiting non--standard asymptotics, problems where estimators converge at rates different from the usual rate (square root of the sample size) and/or have non--Gaussian limit distributions. The two core areas of investigation are: (A) Estimation of functions under shape restrictions, and (B) Estimation of an appropriate ``threshold'' in the domain of a function where sharp and potentially substantial changes occur. Part of the emphasis in shape--restricted inference will be on maximum likelihood/least squares based procedures and in particular, the construction of confidence sets for quantities of interest (like a regression function, a hazard function etc.) by inversion of residual sum of squares or likelihood ratio statistics. This is in light of initial exploration for monotone function models where such statistics are seen to exhibit asymptotically pivotal behavior, which facilitates inference, since nuisance parameters need not be estimated from the data. Shape constraints of interest are monotonicity, unimodality and convexity/concavity, and often a combination of such features. Smoothing under shape constraints will also be investigated in some of these problems, since smoothing typically yields faster rates of convergence and provides automatic estimation of derivatives of functions, which are often of interest, for example in economics. On the threshold estimation front, two main areas will be explored. The first is change--point estimation, where there is a jump in the value of the function of interest or in the value of the derivative, with the focus being on design issues: how to design the sampling mechanism, given a fixed budget (of points that can be sampled) so as to entail precise detection of the jump. The second area concerns studying appropriate notions of a threshold for smooth functions that show rapid change over a short domain. One such notion can be formulated in terms of a function with a finite number of discontinuities that is used simply as an approximation, or a working model for the true function. The discontinuities of the best fitting working model provide a natural description of a threshold. Split--point estimation, as this is known, turns out to be radically different from change point estimation, in light of some initial work and will be investigated in more detail.The proposed research will have diverse applications, ranging from disciplines in public health like biomedical studies and epidemiology to topics in the social sciences (economics) and the physical sciences (astronomy). Shape restrictions, for example, show up 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/public health), and problems associated with the detection of dark matter in galaxies, solutions to which can shed light on the future evolution of the universe (astrophysics). The P.I. is actively involved in collaborations with econometricians, epidemologists and astronomers: the research emanating from this grant will therefore have strong interdisciplinary flavor and will address many real scientific questions of interest. On the threshold estimation front, statistical methods for split point detection are of importance, since split points have been used as a measure of threshold by ecologists in the development of pollution control standards. The problems related to change point detection will be applied to detection of stress thresholds in engineering systems. The statistical methodology to be developed will be circulated to the statistical community in academia and industry through free software packages. On the educational front, some of the research material in this proposal will be incorporated in advanced courses at the graduate level and an interdisciplinary seminar series. Some projects will also serve as dissertation topics for Ph.D. advisees and will therefore play an important role in the training of future statisticians.