Khasminskii and Yin This proposal presents a joint effort of researchers at Wayne State University and visiting scientists from Russia. The proposed research project encompasses two parts: the study of singularly perturbed stochastic systems, and nonparametric statistical analysis. Motivated by various problems in production planning of manufacturing systems, the PIs and their Russian colleagues will study singularly perturbed jump-diffusions, and related problems in homogenization. Problems to be investigated include ones where the diffusion component changes in a fast pace and the jump process vary slowly, and where the jump process is the dominating force and the diffusion changes relatively slowly. For the homogenization problem, the emphasis is on the evolution of the probability distribution of fast and slow components of the processes. Concerning the second part of the proposed project, the emphasis is on asymptotically minimax procedures. The problems to be investigated include: studying the asymptotic behavior of the minimax mean square risk for recovering an infinite dimensional vector from noisy data, deriving conditions on the Fisher information matrix under which an asymptotically exact lower bound can be derived for mean square risk for general estimation problems, obtaining lower bounds for density estimation problems, and uniform lower bounds for integrated mean square risk for recovering a smooth regression function from noisy data, finding asymptotically minimax estimators for interpolation problems, investigating filtering problems under non-Gaussian noise, and proving that a linear estimator is asymptotically minimax up to the second order in the Bayesian risk expansion. The proposed project presents an effort of cooperative research between researchers from Wayne State University and outstanding experts from Russia. The results of their research in singularly perturbed stochastic systems and nonparametric statistical estimation problems should be useful for many appl ications in manufacturing systems, data analysis, control and optimization of complex systems. This cooperative research project is expected to foster the establishment of a long-term collaboration among the researchers.
