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

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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.