Mathematical Sciences: Smoothing Parameter Selection and Related Topic

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The statistical part of this project will be to study a number of estimation methods using nonparametric approaches. In particular kernel estimation and related areas of nonparametric and density estimation applications will be studied, and the smoothing parameter selection problem addressed. Problems to be considered include those of discontinuities or rough density functions, selection for ridge regression, and the effect of discretization. The modelling part of this project involves new developments of the theory and applications of Markov models in both discrete and continuous time. The investigator will concentrate on rates of convergence results and on state and history dependent criteria for such stability. These results will be incorporated into a new approach to Markov decisions processes where the state space is extended to include the decision set into the overall Markovian structure. Stability properties of such general processes will then be established in quite complex environments. Probabilistic modelling and statistical estimation techniques encompass two separate parts of the same activity. The first part endeavors to describe, in manageable terms, models of real and often complex systems in such a way that the inherent randomness in the system is properly represented. The second endeavors to use these models, and data gathered to describe real systems, in such a way that one can estimate the parameters that actually describe what is happening in practice. In the modelling part of this project the investigators will study important models in operations research, time series, and finance and economics, with a focus on developing methods for stability of such systems. In the statistics part of the proposal, the investigators will look at a number of computer intensive ways of estimating parameters with a goal of finding methods which give good estimates quickly.