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

拟议的项目涉及方法和推理策略的非参数和半参数问题表现出非标准渐近，问题估计收敛率不同于通常的速率（样本大小的平方根）和/或有非高斯极限分布。研究的两个核心领域是：（A）形状限制下的函数估计，以及（B）在函数域中发生急剧和潜在实质性变化的适当"阈值“估计。形状限制推理的部分重点将是基于最大似然/最小二乘的程序，特别是感兴趣的数量（如回归函数，风险函数等）的置信集的构建。通过残差平方和或似然比统计的反演。这是根据单调函数模型的初步探索，其中这些统计数据被认为表现出渐近枢轴行为，这有利于推断，因为滋扰参数不需要从数据中估计。感兴趣的形状约束是单调性、单峰性和凸性/凸性，并且通常是这些特征的组合。形状约束下的平滑也将在其中一些问题中进行研究，因为平滑通常会产生更快的收敛速度，并提供函数导数的自动估计，这通常是经济学中的兴趣。在阈值估计方面，将探索两个主要领域。第一个是变化点估计，即有关函数的值或导数的值出现跳跃，重点是设计问题：在（可取样点的）固定预算的情况下，如何设计取样机制，以便精确检测跳跃。第二个领域涉及研究适当的概念的阈值平滑功能，显示在一个短域的快速变化。一个这样的概念可以用一个函数来表述，该函数具有有限数量的不连续性，仅用作近似值，或真函数的工作模型。最佳拟合工作模型的不连续性提供了阈值的自然描述。分裂点估计，因为这是已知的，原来是从根本上不同的变化点估计，在一些初步的工作，并将进行调查，在更详细的研究，拟议中的研究将有不同的应用，从公共卫生学科，如生物医学研究和流行病学的主题，在社会科学（经济学）和物理科学（天文学）。例如，形状限制自然地出现在对公司/公司产品的分析（经济学），对疾病或感染风险的研究（生物医学研究/公共卫生），以及与星系中暗物质的检测相关的问题，解决方案可以揭示宇宙的未来演化（天体物理学）。私家侦探积极参与与计量经济学家、天文学家和天文学家的合作：因此，这项赠款产生的研究将具有强烈的跨学科色彩，并将解决许多感兴趣的真实的科学问题。 在阈值估计方面，用于分裂点检测的统计方法是重要的，因为分裂点已被生态学家在污染控制标准的发展中用作阈值的量度。与变点检测相关的问题将应用于工程系统中应力阈值的检测。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，本建议中的一些研究材料将纳入研究生一级的高级课程和一系列跨学科研讨会。一些项目也将作为博士论文的主题。因此，它将在培训未来的统计人员方面发挥重要作用。

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.