On Bayesian "testimation" and its application to wavelet thresholding

TL;DR

This paper develops a Bayesian testimation approach for wavelet thresholding, providing adaptive estimators that are asymptotically minimax over various function spaces, with theoretical guarantees and simulation validation.

Contribution

It extends Bayesian testimation to wavelet-based function estimation, achieving adaptive minimax properties over Besov spaces and derivatives, with practical simulation results.

Findings

Estimators are asymptotically near-minimax and minimax in Besov spaces.

The approach adapts to both sparse and dense signals.

Simulation shows competitive finite-sample performance.

Abstract

We consider the problem of estimating the unknown response function in the Gaussian white noise model. We first utilize the recently developed Bayesian maximum a posteriori "testimation" procedure of Abramovich et al. (2007) for recovering an unknown high-dimensional Gaussian mean vector. The existing results for its upper error bounds over various sparse $l_p$-balls are extended to more general cases. We show that, for a properly chosen prior on the number of non-zero entries of the mean vector, the corresponding adaptive estimator is simultaneously asymptotically minimax in a wide range of sparse and dense $l_p$-balls. The proposed procedure is then applied in a wavelet context to derive adaptive global and level-wise wavelet estimators of the unknown response function in the Gaussian white noise model. These estimators are then proven to be, respectively, asymptotically near-minimax and minimax in a wide range of Besov balls. These results are also extended to the estimation of derivatives of the response function. Simulated examples are conducted to illustrate the performance of the proposed level-wise wavelet estimator in finite sample situations, and to compare it with several existing