DMS 9626108 Sun This research involves the following non-parametric problems: BUMP HUNTING PROBLEMS, where tests of unimodality against multi-modality and related estimators are studied; CONFIDENCE BANDS for the mean response function in generalized linear models; inferences in PROJECTION PURSUIT; and RANDOM FIELDS, where the maxima of Gaussian random fields are studied using large deviations and techniques from differential geometry. All of these problems involve some nontrivial computations besides methodology development and theoretical analysis. The investigator also studies efficient algorithms and applies the techniques to air pollution data and other high dimensional and categorical data. Modes or bumps in a density estimate of a data set may reveal interesting structures of the data. For example, in evolutionary biology they may indicate that a certain kind of species will be a dominating factor in stabilizing "natural selection", and in high-energy physics they may show the evidence of "partial-wave scattering amplitudes" or even a new particle. BUMP HUNTING techniques studied in this research help decide whether the modes or bumps are real features of the underline distribution of the data or simply a result of sampling fluctuation. RANDOM FIELDS and GENERALIZED LINEAR MODELS can be used to model air pollution data and many other data. PROJECTION PURSUIT is a powerful technique that searches for interesting structures of a high dimensional data via lower dimensional projections. Another objective of the research is to determine the amount of confidence that may be attached to an estimate based on such techniques.
