Compound decision theory and empirical Bayes methods

Annals of Statistics

Volumn 31, Issue 2, 2003, Pages 379-390

  Zhang, Cun Hui ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Compound decision theory; Empirical Bayes; Nonparametric inference; Prediction

Indexed keywords

EID: 0037688247     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1051027872     Document Type: Review

References (112)

1. BARRON, A., BIRGÉ, L. and MASSART, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413.
2. 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.
3. BERGER, J. O. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean. Ann. Statist. 8 716-761.
4. BERGER, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
5. BREIMAN, L., FRIEDMAN, J. H., OLSHEN, R. A. and STONE, C. J. (1984). Classification and Regression Trees. Wadsworth, Belmont, CA.
6. BROWN, L. D. (1966). On the admissibility of invariant estimators of one or more location parameters. Ann. Math. Statist. 37 1087-1136.
7. CARLIN, B. P. and LOUIS, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall, New York.
8. CARROLL, R. J. and HALL, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184-1186.
9. CARTER, G. and ROLPH, J. (1974). Empirical Bayes methods applied to estimating fire alarm probabilities. J. Amer. Statist. Assoc. 69 880-885.
10. COGBURN, R. (1965). On the estimation of a multivariate location parameter with squared error loss. In Bernoulli, Bayes, and Laplace Anniversary Volume (J. Neyman and L. Le Cam, eds.) 24-29. Springer, Berlin.
11. COGBURN, R. (1967). Stringent solutions to statistical decision problems. Ann. Math. Statist. 38 447-463.
12. COPAS, J. B. (1969). Compound decisions and empirical Bayes (with discussion). J. Roy. Statist. Soc. Ser. B 31 397-425.
13. COPAS, J. B. (1972). Empirical Bayes methods and the repeated use of a standard. Biometrika 59 349-360.
14. COVER, T. (1968). Learning in pattern recognition. In Methodologies of Pattern Recognition (S. Watanabe, ed.) 111-132. Academic Press, New York.
15. COVER, T. (1991). Universal portfolios. Math. Finance 1 1-29.
16. DATTA, S. (1991). Asymptotic optimality of Bayes compound estimators in compact exponential families. Ann. Statist. 19 354-365.
17. DEELY, J. J. and KRUSE, R. L. (1968). Construction of sequences estimating the mixing distribution. Ann. Math. Statist. 39 286-288.
18. DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1-38.
19. DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
20. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369.
21. EFROMOVICH, S. (1999). Nonparametric Curve Estimation. Springer, New York.
22. EFROMOVICH, S. and PINSKER, M. S. (1984). An adaptive algorithm of nonparametric filtering. Autom. Remote Control 11 58-65.
23. EFRON, B. (1996). Empirical Bayes methods for combining likelihoods. J. Amer. Statist. Assoc. 91 538-550.
24. EFRON, B. (2003). Robbins, empirical Bayes and microarrays. Ann. Statist. 31 366-378.
25. EFRON, B. and MORRIS, C. (1972a). Empirical Bayes on vector observations: An extension of Stein's method. Biometrika 59 335-347.
26. EFRON, B. and MORRIS, C. (1972b). Limiting the risk of Bayes and empirical Bayes estimators. II. The empirical Bayes case. J. Amer. Statist. Assoc. 67 130-139.
27. EFRON, B. and MORRIS, C. (1973a). Stein's estimation rule and its competitors-an empirical Bayes approach. J. Amer. Statist. Assoc. 68 117-130.
28. EFRON, B. and MORRIS, C. (1973b). Combining possibly related estimation problems (with discussion). J. Roy. Statist. Soc. Ser. B 35 379-421.
29. EFRON, B. and MORRIS, C. (1977). Stein's paradox in statistics. Sci. Amer. 236 119-127.
30. EFRON, B., STOREY, J. and TIBSHIRANI, R. (2001). Microarrays, empirical Bayes methods, and false discovery rates. Technical report, Stanford Univ. Available online at http://www-stat.stanford.edu/~tibs/research.html.
31. EFRON, B., TIBSHIRANI, R., STOREY, J. and TUSHER, V. (2001). Empirical Bayes analysis of a microarray experiment. J. Amer. Statist. Assoc. 96 1151-1160.
32. FAN, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272.
33. FAN, J. and GIJBELS, I. (1996). Local Polynomial Modeling and Its Applications. Chapman and Hall, London.
34. FINKELSTEIN, M. O., LEVIN, B. and ROBBINS, H. (1996a). Clinical and prophylactic trials with assured new treatment for those at greater risk. I. A design proposal. Amer. J. Public Health 86 691-695.
35. FINKELSTEIN, M. O., LEVIN, B. and ROBBINS, H. (1996b). Clinical and prophylactic trials with assured new treatment for those at greater risk. II. Examples. Amer. J. Public Health 86 696-705.
36. FOX, R. J. (1970). Estimating the empiric distribution function of certain parameter sequences. Ann. Math. Statist. 41 1845-1852.
37. FRIEDMAN, J. H. (1991). Multivariate adaptive regression splines. Ann. Statist. 19 1-67.
38. GEORGE, E. I. (1986). Minimax multiple shrinkage estimation. Ann. Statist. 14 188-205.
39. GEORGE, E. I. and FOSTER, D. P. (2000). Calibration and empirical Bayes variable selection. Biometrika 87 731-747.
40. GILLILAND, D. C. (1968). Sequential compound estimation. Ann. Math. Statist. 39 1890-1905.
41. GILLILAND, D. C. and HANNAN, J. F. (1969). On an extended compound decision problem. Ann. Math. Statist. 40 1536-1541.
42. GILLILAND, D. C., HANNAN, J. F. and HUANG, J. S. (1976). Asymptotic solutions to the two state component compound decision problem, Bayes versus diffuse priors on proportions. Ann. Statist. 4 1101-1112.
43. GOOD, I. J. (1953). On the population frequencies of species and the estimation of population parameters. Biometrika 40 237-264.
44. HANNAN, J. F. (1957). Approximation to Bayes risk in repeated play. In Contributions to the Theory of Games (M. Dresher and A. W. Tucker, eds.) 3 97-139. Princeton Univ. Press.
45. HANNAN, J. F. and ROBBINS, H. (1955). Asymptotic solutions of the compound decision problem for two completely specified distributions. Ann. Math. Statist. 26 37-51.
46. HANNAN, J. F. and VAN RYZIN, J. R. (1965). Rate of convergence in the compound decision problem for two completely specified distributions. Ann. Math. Statist. 36 1743-1752.
47. HASTIE, T., TIBSHIRANI, R. and FRIEDMAN, J. H. (2001). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York.
48. HOADLEY, B. (1981). The quality management plan (QMP). Bell System Tech. J. 60 215-273.
49. IBRAGIMOV, I. A. and KHASMINSKII, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.
50. JAMES, W. and STEIN, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361-379. Univ. California Press, Berkeley.
51. JOHNS, M. V. (1957). Non-parametric empirical Bayes procedures. Ann. Math. Statist. 28 649-669.
52. JOHNS, M. V. (1961). An empirical Bayes approach to non-parametric two-way classification. In Studies in Item Analysis and Prediction (H. Solomon, ed.) 221-232. Stanford Univ. Press.
53. JOHNS, M. V. and VAN RYZIN, J. R. (1971). Convergence rates for empirical Bayes two-action problems. I. Discrete case. Ann. Math. Statist. 42 1521-1539.
54. JOHNS, M. V. and VAN RYZIN, J. R. (1972). Convergence rates for empirical Bayes two-action problems. II. Continuous case. Ann. Math. Statist. 43 934-947.
55. KAGAN, A. M. (1962). An empirical Bayes approach to the estimation problem. Dokl. Acad. Sci. U.S.S.R. 147 1020-1021.
56. KIEFER, J. and WOLFOWITZ, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Statist. 27 887-906.
57. LEVIN, B., ROBBINS, H. and ZHANG, C.-H. (2002). Mathematical aspects of estimating two treatment effects and a common variance in an assured allocation design. J. Statist. Plann. Inference 108 255-262.
58. LINDSAY, B. G. (1983). The geometry of mixture likelihoods: A general theory. Ann. Statist. 11 86-94.
59. MARITZ, J. S. and LWIN, T. (1989). Empirical Bayes Methods, 2nd ed. Chapman and Hall, London.
60. MARTZ, H. and KRUTCHKOFF, R. G. (1969). Empirical Bayes estimators in a multiple linear regression model. Biometrika 56 367-374.
61. MEEDEN, G. (1972). Some admissible empirical Bayes procedures. Ann. Math. Statist. 43 96-101.
62. MIYASAWA, K. (1961). An empirical Bayes estimator of the mean of a normal population. Bull. Internat. Statist. Inst. 38 181-188.
63. MORRIS, C. N. (1983). Parametric empirical Bayes inference: Theory and applications. J. Amer. Statist. Assoc. 78 47-55.
64. NEYMAN, J. (1962). Two breakthroughs in the theory of statistical decision-making. Rev. Internat. Statist. Inst. 30 11-27.
65. OATEN, A. (1972). Approximation to Bayes risk in compound decision problems. Ann. Math. Statist. 43 1164-1184.
66. O'BRYAN, T. E. (1976). Some empirical Bayes results in the case of component problems with varying sample sizes for discrete exponential families. Ann. Statist. 4 1290-1293.
67. ROBBINS, H. (1951). Asymptotically subminimax solutions of compound decision problems. Proc. Second Berkeley Symp. Math. Statist. Probab. 1 131-148. Univ. California Press, Berkeley.
68. ROBBINS, H. (1956). An empirical Bayes approach to statistics. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 157-163. Univ. California Press, Berkeley.
69. ROBBINS, H. (1962). Some numerical results on a compound decision problem. In Recent Developments in Information and Decision Processes (R. E. Machol and P. Gray, eds.) 52-62. Macmillan, New York.
70. ROBBINS, H. (1963). The empirical Bayes approach to testing statistical hypotheses. Rev. Internat. Statist. Inst. 31 195-208.
71. ROBBINS, H. (1964). The empirical Bayes approach to statistical decision problems. Ann. Math. Statist. 35 1-20.
72. ROBBINS, H. (1968). Estimating the total probability of the unobserved outcomes of an experiment. Ann. Math. Statist. 39 256-257.
73. ROBBINS, H. (1977). Prediction and estimation for the compound Poisson distribution. Proc. Nat. Acad. Sci. U.S.A. 74 2670-2671.
74. ROBBINS, H. (1980a). Estimation and prediction for mixtures of the exponential distribution. Proc. Nat. Acad. Sci. U.S.A. 77 2382-2383.
75. ROBBINS, H. (1980b). An empirical Bayes estimation problem. Proc. Nat. Acad. Sci. U.S.A. 77 6988-6989.
76. ROBBINS, H. (1982). Estimating many variances. In Statistical Decision Theory and Related Topics III (S. S. Gupta, ed.) 2 251-261. Academic Press, New York.
77. ROBBINS, H. (1983). Some thoughts on empirical Bayes estimation. Ann. Statist. 11 713-723.
78. ROBBINS, H. (1985). Linear empirical Bayes estimation of means and variances. Proc. Nat. Acad. Sci. U.S.A. 82 1571-1574.
79. ROBBINS, H. (1988). The u, v method of estimation. In Statistical Decision Theory and Related Topics IV (S. S. Gupta and J. O. Berger, eds.) 1 265-270. Springer, New York.
80. ROBBINS, H. and SAMUEL, E. (1962). Testing statistical hypotheses; the compound approach. In Recent Developments in Information and Decision Processes (R. E. Machol and P. Gray, eds.) 63-70. Macmillan, New York.
81. ROBBINS, H. and ZHANG, C.-H. (1988), Estimating a treatment effect under biased sampling. Proc. Nat. Acad. Sci. U.S.A. 85 3670-3672.
82. ROBBINS, H. and ZHANG, C.-H. (1989). Estimating the superiority of a drug to a placebo when all and only those patients at risk are treated with the drug. Proc. Nat. Acad. Sci. U.S.A. 86 3003-3005.
83. ROBBINS, H. and ZHANG, C.-H. (1991). Estimating a multiplicative treatment effect under biased allocation. Biometrika 78 349-354.
84. ROBBINS, H. and ZHANG, C.-H. (2000). Efficiency of the u, v method of estimation. Proc. Nat. Acad. Sci. U.S.A. 97 12976-12979.
85. RUBIN, D. (1980). Using empirical Bayes techniques in the law school validity studies (with discussion). J. Amer. Statist. Assoc. 75 801-827.
86. RUTHERFORD, J. R. and KRUTCHKOFF, R. G. (1967). The empirical Bayes approach: Estimating the prior distribution. Biometrika 54 326-328.
87. SAMUEL, E. (1963a). Asymptotic solutions of the sequential compound decision problem. Ann. Math. Statist. 34 1079-1094.
88. SAMUEL, E. (1963b). An empirical Bayes approach to the testing of certain parametric hypotheses. Ann. Math. Statist. 34 1370-1385.
89. SAMUEL, E. (1964). Convergence of the losses of certain decision rules for the sequential compound decision problem. Ann. Math. Statist. 35 1606-1621.
90. SAMUEL, E. (1965). Sequential compound estimators. Ann. Math. Statist. 36 879-889.
91. SIMAR, L. (1976). Maximum likelihood estimation of a compound Poisson process. Ann. Statist. 4 1200-1209.
92. SINGH, R. S. (1979). Empirical Bayes estimation in Lebesgue exponential families with rates near the best possible rate. Ann. Statist. 7 890-902.
93. STEIN, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197-206. Univ. California Press, Berkeley.
94. STONE, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118-171.
95. STRAWDERMAN, W. E. (1971). Proper Bayes minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385-388.
96. SUSARLA, V. (1974). Rate of convergence in the sequence-compound squared-distance loss estimation problem for a family of m-variate normal distributions. Ann. Statist. 2 118-133.
97. SUSARLA, V. and VAN RYZIN, J. R. (1978). Empirical Bayes estimation of a distribution (survival) function from right-censored observations. Ann. Statist. 6 740-754.
98. TEICHER, H. (1961). Identifiability of mixtures. Ann. Math. Statist. 32 244-248.
99. TEICHER, H. (1963). Identifiability of finite mixtures. Ann. Math. Statist. 34 1265-1269.
100. VAN HOUWELINGEN, H. C. and THOROGOOD, J. (1995). Construction, validation and updating of a prognostic model for kidney graft survival. Statistics in Medicine 14 1999-2008.
101. VAN HOUWELINGEN, J. C. (1977). Monotonizing empirical Bayes estimators for a class of discrete distributions with monotone likelihood ratio. Statist. Neerlandica 31 95-104.
102. VAN RYZIN, J. R. (1966a). The sequential compound decision problem with m × n finite loss matrix. Ann. Math. Statist. 37 954-975.
103. VAN RYZIN, J. R. (1966b). Repetitive play in finite statistical games with unknown distributions. Ann. Math. Statist. 37 976-994.
104. VAN RYZIN, J. R. and SUSARLA, V. (1977). On the empirical Bayes approach to multiple decision problems. Ann. Statist. 5 172-181.
105. VARDEMAN, S. B. (1978). Admissible solutions of finite state sequence compound decision problems. Ann. Statist. 6 673-679.
106. WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
107. WIND, S. L. (1973). An empirical Bayes approach to multiple linear regression. Ann. Statist. 1 93-103.
108. ZASLAVSKY, A. M. (1993). Combining census, dual-system, and evaluation study data to estimate population shares. J. Amer. Statist. Assoc. 88 1092-1105.
109. ZHANG, C.-H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-831.
110. ZHANG, C.-H. (1995). On estimating mixing densities in discrete exponential family models. Ann. Statist. 23 929-945.
111. ZHANG, C.-H. (1997). Empirical Bayes and compound estimation of normal means. Statist. Sinica 7 181-193.
112. ZHANG, C.-H. (2000). General empirical Bayes wavelet methods. Technical Report 2000-007, Dept. Statist., Rutgers Univ., Piscataway, NJ.