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On optimality of Bayesian testimation in the normal means problem. (English) Zbl 1126.62003
Summary: We consider the problem of recovering a high-dimensional vector \(\mu \) observed in white noise, where the unknown vector \(\mu \) is assumed to be sparse. The objective of the paper is to develop a Bayesian formalism which gives rise to a family of \(l_{0}\)-type penalties. The penalties are associated with various choices of the prior distributions \(\pi_n(\cdot )\) on the number of nonzero entries of \(\mu \) and, hence, are easy to interpret. The resulting Bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of \(\pi_n(\cdot )\). Furthermore, they achieve optimality in a rather general setting under very mild conditions on the prior. We also specify the class of priors \(\pi_n(\cdot )\) for which the resulting estimator is adaptively optimal (in the minimax sense) for a wide range of sparse sequences and consider several examples of such priors.

62C10 Bayesian problems; characterization of Bayes procedures
62F03 Parametric hypothesis testing
62C20 Minimax procedures in statistical decision theory
62G05 Nonparametric estimation
62F10 Point estimation
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[1] Abramovich, F. and Angelini, C. (2006). Bayesian maximum a posteriori multiple testing procedure. Sankhyā 68 436–460. · Zbl 1193.62031
[2] Abramovich, F. and Benjamini, Y. (1995). Thresholding of wavelet coefficients as a multiple hypotheses testing procedure. In Wavelets and Statistics . Lecture Notes in Statist. 103 5–14. Springer, New York. · Zbl 0875.62081
[3] Abramovich, F. and Benjamini, Y. (1996). Adaptive thresholding of wavelet coefficients. Comput. Statist. Data Anal. 22 351–361.
[4] Abramovich, F., Benjamini, Y., Donoho, D. L. and Johnstone, I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist. 34 584–653. · Zbl 1092.62005
[5] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (B. N. Petrov and F. Csáki, eds.) 267–281. Akadémiai Kiadó, Budapest. · Zbl 0283.62006
[6] Antoniadis, A. and Fan, J. (2001). Regularization of wavelet approximations (with discussion). J. Amer. Statist. Assoc. 96 939–967. JSTOR: · Zbl 1072.62561
[7] 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. JSTOR: · Zbl 0809.62014
[8] Birgé, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc. 3 203–268. · Zbl 1037.62001
[9] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425–455. JSTOR: · Zbl 0815.62019
[10] 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. · Zbl 0802.62006
[11] Donoho, D. L. and Johnstone, I. M. (1996). Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli 2 39–62. · Zbl 0877.62035
[12] 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. JSTOR: · Zbl 0788.62103
[13] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360. JSTOR: · Zbl 1073.62547
[14] Foster, D. and George, E. (1994). The risk inflation criterion for multiple regression. Ann. Statist. 22 1947–1975. · Zbl 0829.62066
[15] Foster, D. and Stine, R. (1999). Local asymptotic coding and the minimum description length. IEEE Trans. Inform. Theory 45 1289–1293. · Zbl 0959.62006
[16] Frank, I. E. and Friedman, J. H. (1993). A statistical view of some chemometrics regression tools (with discussion). Technometrics 35 109–148. · Zbl 0775.62288
[17] Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75 800–802. JSTOR: · Zbl 0661.62067
[18] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scand. J. Statist. 6 65–70. · Zbl 0402.62058
[19] Hunter, D. R. and Li, R. (2005). Variable selection using MM algorithms. Ann. Statist. 33 1617–1642. · Zbl 1078.62028
[20] Johnstone, I. M. (1994). Minimax Bayes, asymptotic minimax and sparse wavelet priors. In Statistical Decision Theory and Related Topics V (S. Gupta and J. Berger, eds.) 303–326. Springer, New York. · Zbl 0815.62017
[21] Johnstone, I. M. (2002). Function estimation and Gaussian sequence models. Unpublished manuscript. · Zbl 1037.91527
[22] Johnstone, I. M. and Silverman, B. W. (2004). Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. Ann. Statist. 32 1594–1649. · Zbl 1047.62008
[23] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257. · Zbl 1101.62349
[24] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464. · Zbl 0379.62005
[25] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288. JSTOR: · Zbl 0850.62538
[26] Tibshirani, R. and Knight, K. (1999). The covariance inflation criterion for adaptive model selection. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 529–546. JSTOR: · Zbl 0924.62031
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