A Study of Boundary Phenomena in a Class of Parametric and Nonparametric Problems

该项目处理一类参数和非参数模型中的边界效应。主要有三个方向。第一个涉及使用基于似然的方法和最小二乘类型标准来估计协变量空间（回归模型）边界的方法和推理技术的发展，其动机主要是：(a)需要建立随机模型作为响应——协变量研究和生存分析，以迎合癌症患者个性化治疗疗法的发展；(b)自适应多阶段策略的发展，用于估计回归表面的不连续。第二个方向涉及欧几里得空间中被封闭曲面包围的区域的确定，其中函数取极值（最大值或最小值）。基于在函数域中不同点进行的统计检验获得的P值，假设函数在这些点上假设其极值，为此开发了一种新的程序。最后一个方向是在单调性或凸性等形状约束下的非参数回归中，量化协变量抽样分布对回归函数的非参数最大似然值的影响。支持协变量的网格的分辨率与这些形状约束估计量的渐近分布密切相关。在这里，人们遇到边界效应，因为存在特定的分辨率，在这个分辨率下，渐近性从高斯到非高斯急剧转变。拟议的研究计划是由几个不同领域的紧迫问题驱动的：从癌症患者的临床试验、流行病学研究和复杂“组学”实验的数据到系统工程、信号处理和功能磁共振成像研究中的问题。在科学方面，基于这项拨款的研究将导致新的基于设计的适应性程序，用于研究工程系统在不同投入强度下的行为，以及临床试验的新设计，以确定涉及癌症进展的核心因素，以及癌症研究中个性化治疗疗法的发展。通过准确识别从FMRI获得的信号来更好地理解大脑激活机制，以及通过传感器网络处理信号的方法。从我们的研究中产生的新方法程序将通过跨学科的互动和合作，以及在易于访问的语言环境中开发软件，传播到相关的科学界。最后，在教育方面，拟议研究的材料将为研究生提供论文主题，研究生也将获得这项资助；因此，该项目将在培训未来统计人员方面发挥重要作用。

The project deals with boundary effects in a class of parametric and nonparametric models. Three main directions are pursued. The first involves thedevelopment of methodological and inferential techniques for estimating boundaries in the covariate space (in regression models) using both likelihood based methods and least squares type criteria and are motivated primarily by (a) the need to build stochastic models in response--covariate studies and survival analysis that cater to the development of individualized treatment therapies for cancer-afflicted individuals, and, (b) the development of adaptive multistage strategies for estimating discontinuities in regression surfaces. The second direction involves the determination of a region enclosed by a closed surface in Euclidean space where a function assumes an extremal value (a maximum or a minimum). A novel procedure based on P--values obtained from statistical tests conducted at different points in the domain of the function, for the hypothesis that the function assumes its extreme value at those points, is developed to this end. The final direction deals with quantifying the effect of the sampling distribution of covariates on nonparametric MLEs of the regression function, in nonparametric regression under shape--constraints like monotonicity or convexity. The resolution of the grid on which the covariate is supported is seen to be critically related to the asymptotic distributions of these shape-constrained estimators. Here, one encounters boundary effects in the sense that there exist specific resolutions at which the asymptotics transition dramatically from Gaussian to non-Gaussian.The proposed research program is motivated by compelling problems in several different areas: from clinical trials for cancer patients, epidemiological studies and data from complex`omics' experiments to problems in systems engineering, signal processing and FMRI studies. On the scientific front, the research based on this grant will lead to new design-based adaptive procedures for studying the behavior of engineering systems under different input intensities as well as new designs for clinical trials to identify core factors involved in cancer progression, to the development of indvidualized treatment therapies in cancer research, to a better understanding of brain activation mechanisms via accurate identification of signals obtained from FMRI and to methods for signal processing via sensor networks. The novel methodological procedures ensuing from our research will be disseminated to the relevant scientific communities, both via inter--disciplinary interaction and collaboration, and the development of software in a readily accessible language environment. Finally, on the educational front, the material from the proposed research will provide dissertation topics for graduate students who will also be supported on this grant; the project will therefore play an important role in the training of future statisticians.