Problems Related to Gaussian Processes

Colorado State University, Fort Collins CO

Investigators

Abstract

Abstract: The investigator studies research problems in theoretical probability. In particular he investigates the problem called small deviations. The PI is also working on extreme value theory for Gaussian random fields motivated by an application to nonparametric statistics. The problem of small deviations is related to the behavior of the probability that a stochastic process stays in a small neighborhood of the origin. As the neighborhood shrinks, this probability clearly tends to zero --- the question is at what rate? The PI calculates this rate for various processes, including the storage process that is used to model the amount of water available in a dam. Motivation of the other problem comes from nonparametric statistics. One of the main problem in modern nonparametric smoothing is concerned with finding the best smooth curve approximating the data. Chaudhuri & Marron (1999) proposed a tool for data exploration called SiZer. The visual display of SiZer, can be viewed as a summary of a large number (hundreds) of hypothesis test results. For reasonable statistical inference, care needs to be taken about the multiple comparison issue. The current implementations of SiZer is not addressing the issue of multiple testing adequately. To correct this one needs to study the distribution of the maximum of a particular discrete nonstationary Gaussian random field. The PI works on two major problems, small deviations for stochastic processes and extreme value theory for a particular random field. The area of small deviations is relatively new with major developments starting in the 1990's. The solution of the small deviation problems helps us understand the nature of certain rare events when the variability of a random process is much less then expected. Applications of this theory are currently pursued by many researchers. The investigator develops a new application of small deviation techniques in storage theory. The proposed extreme value problem is directly motivated by a statistical application. It aims at improving the performance of the data analysis tool SiZer. SiZer is currently widely used by many applied scientists. Successful applications of SiZer include internet traffic modeling, climatology and environment, and biology and genetics. Part of this proposal aims at significantly improving the SiZer tool to make it more reliable for applications.