Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spect...

General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and square L 2 losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under...

In this article we study function estimation via wavelet shrinkage for data with long-range dependence. We propose a fractional Gaussian noise model to approximate nonparametric regression with long-range dependence and establish asymp-totics for minimax risks. Because of long-range dependence, the minimax risk and the minimax linear risk converge to zero at rates that diier from those for data...

This paper studies the issue of optimal deconvolution density estimation using wavelets. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coe cients. Minimax rates of convergence under the integrated square loss are studied over Besov classes B pq of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of converge...

Large data sets often require performing distributed statistical estimation, with a full data set split across multiple machines and limited communication between machines. To study such scenarios, we define and study some refinements of the classical minimax risk that apply to distributed settings, comparing to the performance of estimators with access to the entire data. Lower bounds on these...

This paper focuses on recursive estimation of locally stationary autoregressive processes. The stability of the model is revisited and uniform results are provided when the time-varying autoregression parameters belong to appropiate smoothness classes. An adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations...