Source: Ann. Statist.
Volume 32, Number 4
An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density γ, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are “nearly black” sequences where only a proportion η is allowed to be nonzero, and sequences with normalized ℓp norm bounded by η, for η>0 and 0<p≤2. Estimation error is measured by mean qth power loss, for 0<q≤2. For all the classes considered, and for all q in (0,2], the method achieves the optimal estimation rate as n→∞ and η→0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q>1. Simulations show excellent performance. For appropriately chosen functions γ, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.
Abramovich, F., Benjamini, Y., Donoho, D. L. and Johnstone, I. M. (2000). Adapting to unknown sparsity by controlling the false discovery rate. Technical Report 2000-19, Dept. Statistics, Stanford Univ.
Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115--129.
Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289--300.
Brown, L. D., Johnstone, I. M. and MacGibbon, K. B. (1981). Variation diminishing transformations: A direct approach to total positivity and its statistical applications. J. Amer. Statist. Assoc. 76 824--832.
Mathematical Reviews (MathSciNet): MR650893
Bruce, A. and Gao, H.-Y. (1996). Applied Wavelet Analysis with S-PLUS. Springer, New York.
Cai, T. T. (2002). On block thresholding in wavelet regression: Adaptivity, block size, and threshold level. Statist. Sinica 12 1241--1273.
Cai, T. T. and Silverman, B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation. Sankhyā Ser. B 63 127--148.
Donoho, D. L. and Johnstone, I. M. (1994). Minimax risk over $\ell_p$-balls for $\ell_q$-error. Probab. Theory Related Fields 99 277--303.
Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200--1224.
Donoho, D. L., Johnstone, I. M., Hoch, J. C. and Stern, A. S. (1992). Maximum entropy and the nearly black object (with discussion). J. Roy. Statist. Soc. Ser. B 54 41--81.
George, E. I. and Foster, D. P. (1998). Empirical Bayes variable selection. In Proc. Workshop on Model Selection, Special Issue of Rassegna di Metodi Statistici ed Applicazioni (W. Racugno, ed.) 79--108. Pitagora Editrice, Bologna.
George, E. I. and Foster, D. P. (2000). Calibration and empirical Bayes variable selection. Biometrika 87 731--747.
Johnstone, I. M. and Silverman, B. W. (2003). EbayesThresh: R and S-PLUS software for Empirical Bayes thresholding. Available at www.stats.ox.ac.uk/~silverma/ebayesthresh.
Johnstone, I. M. and Silverman, B. W. (2004). Empirical Bayes selection of wavelet thresholds. Ann. Statist. To appear.
Karlin, S. (1968). Total Positivity 1. Stanford Univ. Press, Stanford, CA.
Mathematical Reviews (MathSciNet): MR230102
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
Mathematical Reviews (MathSciNet): MR762984
Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135--1151.
Mathematical Reviews (MathSciNet): MR630098
Zhang, C.-H. (2004). General empirical Bayes wavelet methods and exactly adaptive minimax estimation. Ann. Statist. To appear.