Markov process techniques and asymptotics for the analysis of spatial and temporal data

该计划将重点结合现代统计学的创新技术，包括纵向数据分析和贝叶斯非参数统计，以及概率论和随机过程的基础研究。a)广义估计方程（GEE）方法是广义线性模型的推广，为纵向数据分析提供了一种半参数方法。当研究中缺少一些响应时，该程序将集中于与回归参数的一致性和渐近正态性相关的问题。在这种情况下使用经验似然法构成了一种新颖的方法，它有可能产生可应用于广泛数据情况的新结果。b)在所有分布函数的空间上构造先验分布，这些分布函数具有很大的支持和分析上可处理的后验，是现代统计学中许多研究的核心。基于这一领域的最新进展，本课程将重点关注与马尔可夫先验的后验一致性相关的问题，以及它在审查数据中的应用。c)研究计划的这一部分将集中于与某些类别的自归一化观测相关的各种统计量的渐近行为，这些观测可能描述依赖于许多参数的系统的演化。这将对基础研究这一前景广阔的新领域作出重要贡献。d)随机偏微分方程（SPDE）的解通常是一个在时间和空间上表现随机的过程。在许多应用中，了解解过程何时具有胚马尔可夫性质是很重要的，这要求在给定的时空区域内，该过程的“未来”演化独立于它的过去给定的当前状态。这个程序的目标是研究一类很大的SPDE，它们的解过程具有马尔可夫性质。

This program will focus on combining innovative techniques in modern statistics, including the analysis of longitudinal data and Bayesian nonparametric statistics, with fundamental research in probability theory and stochastic processes. a) The generalized estimation equation (GEE) approach is an extension of generalized linear models, which provides a semiparametric approach to longitudinal data analysis. This program will concentrate on issues related to the consistency and asymptotic normality of the regression parameter, when some of the responses are missing from the study. The use of the empirical likelihood method in this context constitutes a novel approach, which has the potential to generate new results that can be applied to a broad range of data situations. b) The construction of prior distributions on the space of all distribution functions, which have large support and analytically tractable posteriors is at the core of many investigations in modern statistics. Building on recent advances in this area, the present program will concentrate on issues related to the posterior consistency of the Markov prior, as well as its applications to censored data. c) This part of the research program will focus on the asymptotic behavior of various statistics associated to certain classes of self-normalized observations, which may describe the evolution of systems depending on many parameters. This will constitute an important contribution in a promising new area of fundamental research. d) The solution of a stochastic partial differential equation (SPDE) is typically a process that behaves randomly in time and space. In many applications, it is important to understand when the solution process possesses the germ Markov property, which requires that the "future" evolution of the process be independent of its past given the present status, for a given time-space region. The goal of this program will be to examine a large class of SPDE's whose solution processes possess the germ Markov property.