DMS 9625412 Efromovich This research on curve estimation focuses on optimal adaptive nonparametric time series estimators that are: (i) asymptotically efficient for different loss functions, (ii) well performing for the case of small sample sizes in comparison with peer oracles based on underlying curve, (iii) efficiently data-compressing for problems arising in environmental problems like monitoring quality of water or the analysis of global change via marine magnetic anomaly; (iv) robust to distribution of noise and long covariance observations. The asymptotic analysis is based on the study of local empirical processes, modern probabilistic results for mixing sequences and sharp data-driven estimators of spectral density. Data-driven estimators for the case of small sample sizes are explored both theoretically via oracle inequalities and numerically via intensive Monte Carlo study. %%% The research involves the development of optimal methods for the recovery of images that arise in different scientific problems including: (i) secretion of hormones such as insulin secretion where no other methods have been successful so far due to noisy, long-dependent and indirect data; (ii) monitoring quality of water based on the analysis of large, correlated and sparse data sets that have undergone efficient, computationally intensive data-compression; (iii) timing of geological events, including plate motions and climatological variations; (iv) testing and modeling properties of new materials. ***
