My research interests lie primarily in the areas of applied probability, time series, and stochastic processes. My dissertation work focused on extreme values of general stationary processes. While my research interests have gravitated towards problems in time series analysis (inference, estimation, prediction and general properties of time series models), extreme value theory still has a strong influence in my approach to solving problems.
In the 1980s, Sid Resnick and I wrote a series of papers developing a theory of extremes for a wide range of time series models, including linear models and special cases of bilinear models. Point process techniques, which played a key role in much of this work, were exploited to establish limit theory for a variety of nonextreme-based statistics such as the sample mean, the sample autocovariance function, and the sample autocorrelation function, for heavy-tailed data. More recently, these ideas were extended by Mikosch and myself to nonlinear time series models that are often used for the analysis of financial data. Generalized autoregressive conditional hetereoscedastic (GARCH) models and stochastic volatility models are examples of processes that fall under this general theory.
NonGaussian linear models and nonlinear time series models have also been frequent objects of study in my research. In the former, we have considered estimation problems in nonstandard situations such as non-casual and/or non-invertible ARMA models and moving averages with unit roots. On the nonlinear side, we have studied properties of financial time series models and have proposed models for analyzing time series of count data. Spatial-modeling with application to environmental problems is an emerging research theme of mine. While many of the ideas used in the time series setting carry over to this new setting, spatial data offers a new set of modeling challenges. Together with colleagues at CSU, we have formed the Space-Time Aquatic Resources Modeling and Analysis Program (STARMAP) with the support of an EPA-STAR grant.