Asymptotically Minimax Adaptive Estimation. I: Upper Bounds. Optimally Adaptive Estimates
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The author presents some new solutions of functional adaptive estimation problems arising in stochastic systems with disturbing parameters affecting the accuracy of estimation. The problems considered include estimation of a signal in a Gaussian white noise, estimation of a functional acting on such a signal, prediction in a polynomial regression with unknown order of the polynomial and so on. The author answers the following questions related to these problems: 1. What are sufficient conditions for existence of optimally adaptive estimates? 2. What are general rules for optimally adaptive estimators? 3. How to use general results to design optimally adaptive estimators of signals in \(L^ p\)- and \(C\)-spaces and their functionals for observations in Gaussian white noise? 4. How to find sufficient conditions for nonexistence of optimally adaptive estimators? The rigorous treatment in the paper makes it difficult to read but leads to much stronger results than obtained by other authors for similar problems [see e.g. \textit{W. Härdle} and \textit{J. S. Marron}, Ann. Stat. 13, 1465-1481 (1985; Zbl 0594.62043)].
sufficient conditions, Estimation and detection in stochastic control theory, stochastic systems, accuracy of estimation, Gaussian processes, prediction, Gaussian white noise, Density estimation, upper bounds, functional adaptive estimation problems, disturbing parameters, polynomial regression, asymptotically minimax adaptive estimation, signal estimation, existence of optimally adaptive estimates, nonexistence of optimally adaptive estimators
sufficient conditions, Estimation and detection in stochastic control theory, stochastic systems, accuracy of estimation, Gaussian processes, prediction, Gaussian white noise, Density estimation, upper bounds, functional adaptive estimation problems, disturbing parameters, polynomial regression, asymptotically minimax adaptive estimation, signal estimation, existence of optimally adaptive estimates, nonexistence of optimally adaptive estimators
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