Extreme Values, Time Series and Prediction

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The research in this project focuses on a number of theoretical and applied topics concerned with prediction and extreme values in time series. Theoretical research includes: (1) spectral methods for fitting regression models where the residuals form a bivariate time series with unknown spectral density of possibly long-range dependent form; (2) extensions of the methodology to spatial-temporal series; (3) decision-theoretic comparisons of Bayesian and frequentist methods for prediction of extreme values, and (4) extensions of (3) to multivariate processes, including ideas both from hierarchical models and multivariate extreme value theory. The applications which have motivated these topics come from two distinct areas, namely (a) climatology, and (b) the evaluation of risk in insurance and finance. On (a), the P.I. has previously collaborated with two leading climatologists (T. Wigley and B. Santer) on an autocorrelation-based method of detecting climate signals in series of hemispheric temperature averages. The new research, which continues this collaboration, includes more extensive and rigorous techniques for separating signals from background noise in cases where the statistical structure of the background noise is unknown. Extensions will be developed to cases in which the patterns of temperature over the earth's surface are described, not just through northern and Southern Hemisphere averages, but through much more detailed summaries over spatial grid boxes. The benefit of this research is improved methods for detecting anthropogenic influence on the earth's climate, which related to the Federal strategic interest in Environment and Global Change (EGCH). The second topic (b) is concerned with extensions of the idea of "value at risk" which has become a widely studied topic in the fields of insurance and finance. Value at risk is a collection of tools for assessing the probability of very large financial losses, but most currently available systems do not use the most advanced statistical techniques available. The new research will assess alternative measures of risk using statistical methods derived from the mathematical theory of extreme value probabilities.