Wavelet shrinkage is an increasingly popular method for data compression and denoising. Bayesian methods, which offer coherent data-dependent shrinkage, have exhibited excellent mean squared error (MSE) properties in several studies. However, elicitation of the prior distribution hyperparameters is a difficult task. In this paper, we propose an Empirical Bayes (EB) approach for estimating all hyperparameters of the prior distribution for a conjugate analysis, that includes normal error models, but also includes heavier tailed error distributions such as the Student $t$. Using the EB prior distributions, we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging. These EB estimators are computationally competitive with standard classical thresholding methods and are robust to outliers in both the data and wavelet domains. Simulated and real examples are used to illustrate the flexibility and improved MSE performance of these methods in a wide variety of settings.

KEYWORDS: BAYESIAN MODEL AVERAGING; EM; HIERARCHICAL MODELS; MODEL SELECTION; MULTIPLE SHRINKAGE; OUTLIERS; ORTHOGONAL REGRESSION; ROBUSTNESS; THRESHOLDING.

Revised March 2000

Merlise Clyde is Assistant Professor of Statistics, Institute of Statistics and Decision Sciences, Duke University, Durham, NC, 27708-0251, clyde@isds.duke.edu. Edward I. George is the Ed and Molly Smith Chair of Business Administration and Professor of Statistics, Department of MSIS, University of Texas, Austin, TX 78712-1175, egeorge@mail.utexas.edu. This work was supported by NSF grants DMS-96.26135, DMS-97.33013, and DMS-98.03756, and Texas ARP grants 003658.130 and 003658.452.

The manuscript is available in PostScript and PDF formats.