Mathematical Sciences: Inference for Nonlinear Time Series and Spatial Models

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This proposal is concerned with problems related to statistical inference in nonlinear time series, state-space models, random fields and spatial point processes. The emphasis will be on nonlinear and non-Gaussian models. A unified approach will be used to study the asymptotic efficiency properties of the estimators and tests. The local asymptotic normality will be investigated for the likelihood-based models. the quasi- likelihood framework will be used for the cases where the likelihood function is unavailable. Empirical Bayes methods will also be investigated. The statistical analysis and the models to be studied in this proposal will have an impact on a large number of diverse applications in Basic Sciences, Engineering, Health Sciences, Economic and Social Planning, among other. Statistical data such as the unemployment figures, various economic indicators, stock market fluctuations, opinion polls, currency exchange rates and the like are all subject to inherent random variations, and possibly trends. These aspects involving uncertainty can usefully be represented by appropriate probability models such as the ones proposed here. Various methods of forecasting and assessing the precision of the forecasts will also be studied. Techniques of statistical estimation and methods of testing the validity of the models proposed will be investigated.