Compound decision theory and empirical Bayes methods. (English) Zbl 1039.62005
From the introduction: Compound decision theory and empirical Bayes methodology, acclaimed as “two breakthroughs” by J. Neyman [Rev. Inst. Int. Stat. 30, 11–27 (1962; Zbl 0131.35601)], are the most important contributions of Herbert Robbins to statistics. The purpose of this paper is to provide a brief description of his work in these two intimately connected fields, its impact and a number of important related developments.
H. Robbins introduced compound decision theory in 1950 [Proc. Berkeley Symp. Math. Stat. Probab., 1950, 131–148 (1951; Zbl 0044.14803)]. Compound decision theory concerns a sequence of independent statistical decision problems of the same form. Its basic thrust is the possibility of gaining substantial reduction of total risk by allowing statistical procedures for the individual component problems to depend on the observations in the entire sequence. It demonstrates, against naive intuition, that stochastically independent experiments are not necessarily “noninformative” to each other in statistical decision making.
Five years later, H. Robbins [Proc. 3rd Berkeley Symp. Math. Stat. Probab. 1, 157–163 (1956; Zbl 0074.35302)] developed empirical Bayes (EB) theory. EB concerns experiments in which the unknown parameters are i.i.d. random variables with an unknown common prior distribution. EB methodologies provide statistical procedures which approximate the ideal Bayes rule for the true model, so that the goal of the Bayesian inference is nearly achieved without specifying a prior. EB procedures usually perform well conditionally on the unknown parameters and thus provide solutions to compound decision problems. EB methods also find applications in problems with more complex structures and for inference about multivariate and infinite-dimensional parameters in a single experiment.
H. Robbins introduced compound decision theory in 1950 [Proc. Berkeley Symp. Math. Stat. Probab., 1950, 131–148 (1951; Zbl 0044.14803)]. Compound decision theory concerns a sequence of independent statistical decision problems of the same form. Its basic thrust is the possibility of gaining substantial reduction of total risk by allowing statistical procedures for the individual component problems to depend on the observations in the entire sequence. It demonstrates, against naive intuition, that stochastically independent experiments are not necessarily “noninformative” to each other in statistical decision making.
Five years later, H. Robbins [Proc. 3rd Berkeley Symp. Math. Stat. Probab. 1, 157–163 (1956; Zbl 0074.35302)] developed empirical Bayes (EB) theory. EB concerns experiments in which the unknown parameters are i.i.d. random variables with an unknown common prior distribution. EB methodologies provide statistical procedures which approximate the ideal Bayes rule for the true model, so that the goal of the Bayesian inference is nearly achieved without specifying a prior. EB procedures usually perform well conditionally on the unknown parameters and thus provide solutions to compound decision problems. EB methods also find applications in problems with more complex structures and for inference about multivariate and infinite-dimensional parameters in a single experiment.
MSC:
| 62C12 | Empirical decision procedures; empirical Bayes procedures |
| 62C25 | Compound decision problems in statistical decision theory |
| 62G08 | Nonparametric regression and quantile regression |
| 62G05 | Nonparametric estimation |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Software:
ElemStatLearnReferences:
| [1] | BARRON, A., BIRGÉ, L. and MASSART, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413. · Zbl 0946.62036 · doi:10.1007/s004400050210 |
| [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. JSTOR: · Zbl 0809.62014 |
| [3] | BERGER, J. O. (1980). A robust generalized Bay es estimator and confidence region for a multivariate normal mean. Ann. Statist. 8 716-761. · Zbl 0464.62026 · doi:10.1214/aos/1176345068 |
| [4] | BERGER, J. O. (1985). Statistical Decision Theory and Bayesian Analy sis, 2nd ed. Springer, New York. · Zbl 0572.62008 |
| [5] | BREIMAN, L., FRIEDMAN, J. H., OLSHEN, R. A. and STONE, C. J. (1984). Classification and Regression Trees. Wadsworth, Belmont, CA. · Zbl 0541.62042 |
| [6] | BROWN, L. D. (1966). On the admissibility of invariant estimators of one or more location parameters. Ann. Math. Statist. 37 1087-1136. · Zbl 0156.39401 · doi:10.1214/aoms/1177699259 |
| [7] | CARLIN, B. P. and LOUIS, T. A. (1996). Bay es and Empirical Bay es Methods for Data Analy sis. 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. JSTOR: · Zbl 0673.62033 · doi:10.2307/2290153 |
| [9] | CARTER, G. and ROLPH, J. (1974). Empirical Bay es 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, Bay es, and Laplace Anniversary Volume (J. Ney man and L. Le Cam, eds.) 24-29. Springer, Berlin. · Zbl 0147.18404 |
| [11] | COGBURN, R. (1967). Stringent solutions to statistical decision problems. Ann. Math. Statist. 38 447-463. · Zbl 0155.25203 · doi:10.1214/aoms/1177698960 |
| [12] | COPAS, J. B. (1969). Compound decisions and empirical Bay es (with discussion). J. Roy. Statist. Soc. Ser. B 31 397-425. JSTOR: · Zbl 0186.51905 |
| [13] | COPAS, J. B. (1972). Empirical Bay es methods and the repeated use of a standard. Biometrika 59 349-360. JSTOR: · Zbl 0238.62007 · doi:10.1093/biomet/59.2.349 |
| [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. · Zbl 0900.90052 · doi:10.1111/j.1467-9965.1991.tb00002.x |
| [16] | DATTA, S. (1991). Asy mptotic optimality of Bay es compound estimators in compact exponential families. Ann. Statist. 19 354-365. · Zbl 0741.62004 · doi:10.1214/aos/1176347987 |
| [17] | DEELY, J. J. and KRUSE, R. L. (1968). Construction of sequences estimating the mixing distribution. Ann. Math. Statist. 39 286-288. · Zbl 0174.22302 · doi:10.1214/aoms/1177698536 |
| [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. JSTOR: · Zbl 0364.62022 |
| [19] | DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224. JSTOR: · Zbl 0869.62024 · doi:10.2307/2291512 |
| [20] | DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asy mptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR: |
| [21] | EFROMOVICH, S. (1999). Nonparametric Curve Estimation. Springer, New York. · Zbl 0935.62039 |
| [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 Bay es methods for combining likelihoods. J. Amer. Statist. Assoc. 91 538-550. JSTOR: · Zbl 0868.62018 · doi:10.2307/2291646 |
| [24] | EFRON, B. (2003). Robbins, empirical Bay es and microarray s. Ann. Statist. 31 366-378. · Zbl 1038.62099 · doi:10.1214/aos/1051027871 |
| [25] | EFRON, B. and MORRIS, C. (1972a). Empirical Bay es on vector observations: An extension of Stein’s method. Biometrika 59 335-347. JSTOR: · Zbl 0238.62072 · doi:10.1093/biomet/59.2.335 |
| [26] | EFRON, B. and MORRIS, C. (1972b). Limiting the risk of Bay es and empirical Bay es estimators. II. The empirical Bay es case. J. Amer. Statist. Assoc. 67 130-139. JSTOR: · Zbl 0231.62013 · doi:10.2307/2284711 |
| [27] | EFRON, B. and MORRIS, C. (1973a). Stein’s estimation rule and its competitors-an empirical Bay es approach. J. Amer. Statist. Assoc. 68 117-130. JSTOR: · Zbl 0275.62005 · doi:10.2307/2284155 |
| [28] | EFRON, B. and MORRIS, C. (1973b). Combining possibly related estimation problems (with discussion). J. Roy. Statist. Soc. Ser. B 35 379-421. JSTOR: · Zbl 0281.62030 |
| [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). Microarray s, empirical Bay es methods, and false discovery rates. Technical report, Stanford Univ. Available online at http://wwwstat.stanford.edu/ tibs/research.html. URL: |
| [31] | EFRON, B., TIBSHIRANI, R., STOREY, J. and TUSHER, V. (2001). Empirical Bay es analysis of a microarray experiment. J. Amer. Statist. Assoc. 96 1151-1160. JSTOR: · Zbl 1073.62511 · doi:10.1198/016214501753382129 |
| [32] | FAN, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033 · doi:10.1214/aos/1176348248 |
| [33] | FAN, J. and GIJBELS, I. (1996). Local Poly nomial Modeling and Its Applications. Chapman and Hall, London. · Zbl 0873.62037 |
| [34] | FINKELSTEIN, M. O., LEVIN, B. and ROBBINS, H. (1996a). Clinical and prophy lactic 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 prophy lactic 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. · Zbl 0237.62007 · doi:10.1214/aoms/1177696685 |
| [37] | FRIEDMAN, J. H. (1991). Multivariate adaptive regression splines. Ann. Statist. 19 1-67. · Zbl 0765.62064 · doi:10.1214/aos/1176347963 |
| [38] | GEORGE, E. I. (1986). Minimax multiple shrinkage estimation. Ann. Statist. 14 188-205. · Zbl 0602.62041 · doi:10.1214/aos/1176349849 |
| [39] | GEORGE, E. I. and FOSTER, D. P. (2000). Calibration and empirical Bay es variable selection. Biometrika 87 731-747. JSTOR: · Zbl 1029.62008 · doi:10.1093/biomet/87.4.731 |
| [40] | GILLILAND, D. C. (1968). Sequential compound estimation. Ann. Math. Statist. 39 1890-1905. · Zbl 0177.46301 · doi:10.1214/aoms/1177698019 |
| [41] | GILLILAND, D. C. and HANNAN, J. F. (1969). On an extended compound decision problem. Ann. Math. Statist. 40 1536-1541. · Zbl 0183.47803 · doi:10.1214/aoms/1177697371 |
| [42] | GILLILAND, D. C., HANNAN, J. F. and HUANG, J. S. (1976). Asy mptotic solutions to the two state component compound decision problem, Bay es versus diffuse priors on proportions. Ann. Statist. 4 1101-1112. · Zbl 0344.62007 · doi:10.1214/aos/1176343645 |
| [43] | GOOD, I. J. (1953). On the population frequencies of species and the estimation of population parameters. Biometrika 40 237-264. JSTOR: · Zbl 0051.37103 · doi:10.1093/biomet/40.3-4.237 |
| [44] | HANNAN, J. F. (1957). Approximation to Bay es risk in repeated play. In Contributions to the Theory of Games (M. Dresher and A. W. Tucker, eds.) 3 97-139. Princeton Univ. Press. · Zbl 0078.32804 |
| [45] | HANNAN, J. F. and ROBBINS, H. (1955). Asy mptotic solutions of the compound decision problem for two completely specified distributions. Ann. Math. Statist. 26 37-51. · Zbl 0064.38703 · doi:10.1214/aoms/1177728591 |
| [46] | HANNAN, J. F. and VAN Ry ZIN, J. R. (1965). Rate of convergence in the compound decision problem for two completely specified distributions. Ann. Math. Statist. 36 1743-1752. · Zbl 0161.37901 · doi:10.1214/aoms/1177699802 |
| [47] | HASTIE, T., TIBSHIRANI, R. and FRIEDMAN, J. H. (2001). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York. · Zbl 0973.62007 |
| [48] | HOADLEY, B. (1981). The quality management plan (QMP). Bell Sy stem Tech. J. 60 215-273. |
| [49] | IBRAGIMOV, I. A. and KHASMINSKII, R. Z. (1981). Statistical Estimation: Asy mptotic Theory. Springer, New York. |
| [50] | JAMES, W. and STEIN, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Sy mp. Math. Statist. Probab. 1 361-379. Univ. California Press, Berkeley. · Zbl 1281.62026 |
| [51] | JOHNS, M. V. (1957). Non-parametric empirical Bay es procedures. Ann. Math. Statist. 28 649-669. · Zbl 0088.35304 · doi:10.1214/aoms/1177706877 |
| [52] | JOHNS, M. V. (1961). An empirical Bay es approach to non-parametric two-way classification. In Studies in Item Analy sis and Prediction (H. Solomon, ed.) 221-232. Stanford Univ. Press. · Zbl 0106.13001 |
| [53] | JOHNS, M. V. and VAN Ry ZIN, J. R. (1971). Convergence rates for empirical Bay es two-action problems. I. Discrete case. Ann. Math. Statist. 42 1521-1539. · Zbl 0267.62018 · doi:10.1214/aoms/1177693151 |
| [54] | JOHNS, M. V. and VAN Ry ZIN, J. R. (1972). Convergence rates for empirical Bay es two-action problems. II. Continuous case. Ann. Math. Statist. 43 934-947. · Zbl 0249.62014 · doi:10.1214/aoms/1177692557 |
| [55] | KAGAN, A. M. (1962). An empirical Bay es 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. · Zbl 0073.14701 · doi:10.1214/aoms/1177728066 |
| [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. · Zbl 1021.62092 · doi:10.1016/S0378-3758(02)00312-9 |
| [58] | LINDSAY, B. G. (1983). The geometry of mixture likelihoods: A general theory. Ann. Statist. 11 86-94. · Zbl 0512.62005 · doi:10.1214/aos/1176346059 |
| [59] | MARITZ, J. S. and LWIN, T. (1989). Empirical Bay es Methods, 2nd ed. Chapman and Hall, London. · Zbl 0731.62040 |
| [60] | MARTZ, H. and KRUTCHKOFF, R. G. (1969). Empirical Bay es estimators in a multiple linear regression model. Biometrika 56 367-374. · Zbl 0186.52801 · doi:10.1093/biomet/56.2.367 |
| [61] | MEEDEN, G. (1972). Some admissible empirical Bay es procedures. Ann. Math. Statist. 43 96-101. · Zbl 0238.62008 · doi:10.1214/aoms/1177692705 |
| [62] | MIy ASAWA, K. (1961). An empirical Bay es estimator of the mean of a normal population. Bull. Internat. Statist. Inst. 38 181-188. · Zbl 0114.10301 |
| [63] | MORRIS, C. N. (1983). Parametric empirical Bay es inference: Theory and applications. J. Amer. Statist. Assoc. 78 47-55. JSTOR: · Zbl 0506.62005 · doi:10.2307/2287098 |
| [64] | NEy MAN, J. (1962). Two breakthroughs in the theory of statistical decision-making. Rev. Internat. Statist. Inst. 30 11-27. · Zbl 0131.35601 · doi:10.2307/1402069 |
| [65] | OATEN, A. (1972). Approximation to Bay es risk in compound decision problems. Ann. Math. Statist. 43 1164-1184. · Zbl 0241.62005 · doi:10.1214/aoms/1177692469 |
| [66] | O’BRy AN, T. E. (1976). Some empirical Bay es results in the case of component problems with varying sample sizes for discrete exponential families. Ann. Statist. 4 1290-1293. · Zbl 0346.62007 · doi:10.1214/aos/1176343661 |
| [67] | ROBBINS, H. (1951). Asy mptotically subminimax solutions of compound decision problems. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 1 131-148. Univ. California Press, Berkeley. · Zbl 0044.14803 |
| [68] | ROBBINS, H. (1956). An empirical Bay es approach to statistics. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 1 157-163. Univ. California Press, Berkeley. · Zbl 0074.35302 |
| [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 Bay es approach to testing statistical hy potheses. Rev. Internat. Statist. Inst. 31 195-208. · Zbl 0117.14104 · doi:10.2307/1401373 |
| [71] | ROBBINS, H. (1964). The empirical Bay es approach to statistical decision problems. Ann. Math. Statist. 35 1-20. · Zbl 0138.12304 · doi:10.1214/aoms/1177703729 |
| [72] | ROBBINS, H. (1968). Estimating the total probability of the unobserved outcomes of an experiment. Ann. Math. Statist. 39 256-257. · Zbl 0155.25801 · doi:10.1214/aoms/1177698526 |
| [73] | ROBBINS, H. (1977). Prediction and estimation for the compound Poisson distribution. Proc. Nat. Acad. Sci. U.S.A. 74 2670-2671. JSTOR: · Zbl 0359.62014 · doi:10.1073/pnas.74.7.2670 |
| [74] | ROBBINS, H. (1980a). Estimation and prediction for mixtures of the exponential distribution. Proc. Nat. Acad. Sci. U.S.A. 77 2382-2383. JSTOR: · Zbl 0462.62026 · doi:10.1073/pnas.77.5.2382 |
| [75] | ROBBINS, H. (1980b). An empirical Bay es estimation problem. Proc. Nat. Acad. Sci. U.S.A. 77 6988-6989. JSTOR: · Zbl 0456.62029 · doi:10.1073/pnas.77.12.6988 |
| [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. · Zbl 0599.62036 |
| [77] | ROBBINS, H. (1983). Some thoughts on empirical Bay es estimation. Ann. Statist. 11 713-723. · Zbl 0522.62024 · doi:10.1214/aos/1176346239 |
| [78] | ROBBINS, H. (1985). Linear empirical Bay es estimation of means and variances. Proc. Nat. Acad. Sci. U.S.A. 82 1571-1574. JSTOR: · Zbl 0559.62034 · doi:10.1073/pnas.82.6.1571 |
| [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 hy potheses; 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. JSTOR: · Zbl 0639.62098 · doi:10.1073/pnas.85.11.3670 |
| [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. JSTOR: · Zbl 0665.62108 · doi:10.1073/pnas.86.9.3003 |
| [83] | ROBBINS, H. and ZHANG, C.-H. (1991). Estimating a multiplicative treatment effect under biased allocation. Biometrika 78 349-354. JSTOR: · Zbl 0735.62031 · doi:10.1093/biomet/78.2.349 |
| [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. · Zbl 0968.62030 · doi:10.1073/pnas.97.24.12976 |
| [85] | RUBIN, D. (1980). Using empirical Bay es techniques in the law school validity studies (with discussion). J. Amer. Statist. Assoc. 75 801-827. · Zbl 0444.62089 · doi:10.2307/2287648 |
| [86] | RUTHERFORD, J. R. and KRUTCHKOFF, R. G. (1967). The empirical Bay es approach: Estimating the prior distribution. Biometrika 54 326-328. JSTOR: · Zbl 0158.17607 · doi:10.1093/biomet/54.1-2.326 |
| [87] | SAMUEL, E. (1963a). Asy mptotic solutions of the sequential compound decision problem. Ann. Math. Statist. 34 1079-1094. · Zbl 0202.49102 · doi:10.1214/aoms/1177704032 |
| [88] | SAMUEL, E. (1963b). An empirical Bay es approach to the testing of certain parametric hy potheses. Ann. Math. Statist. 34 1370-1385. · Zbl 0202.49304 · doi:10.1214/aoms/1177703870 |
| [89] | SAMUEL, E. (1964). Convergence of the losses of certain decision rules for the sequential compound decision problem. Ann. Math. Statist. 35 1606-1621. · Zbl 0134.35903 · doi:10.1214/aoms/1177700385 |
| [90] | SAMUEL, E. (1965). Sequential compound estimators. Ann. Math. Statist. 36 879-889. · Zbl 0147.17903 · doi:10.1214/aoms/1177700060 |
| [91] | SIMAR, L. (1976). Maximum likelihood estimation of a compound Poisson process. Ann. Statist. 4 1200-1209. · Zbl 0362.62095 · doi:10.1214/aos/1176343651 |
| [92] | SINGH, R. S. (1979). Empirical Bay es estimation in Lebesgue exponential families with rates near the best possible rate. Ann. Statist. 7 890-902. · Zbl 0411.62019 · doi:10.1214/aos/1176344738 |
| [93] | STEIN, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 1 197-206. Univ. California Press, Berkeley. · Zbl 0073.35602 |
| [94] | STONE, C. J. (1994). The use of poly nomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118-171. · Zbl 0827.62038 · doi:10.1214/aos/1176325361 |
| [95] | STRAWDERMAN, W. E. (1971). Proper Bay es minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385-388. · Zbl 0222.62006 · doi:10.1214/aoms/1177693528 |
| [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. · Zbl 0275.62007 · doi:10.1214/aos/1176342618 |
| [97] | SUSARLA, V. and VAN Ry ZIN, J. R. (1978). Empirical Bay es estimation of a distribution (survival) function from right-censored observations. Ann. Statist. 6 740-754. · Zbl 0378.62006 · doi:10.1214/aos/1176344249 |
| [98] | TEICHER, H. (1961). Identifiability of mixtures. Ann. Math. Statist. 32 244-248. · Zbl 0146.39302 · doi:10.1214/aoms/1177705155 |
| [99] | TEICHER, H. (1963). Identifiability of finite mixtures. Ann. Math. Statist. 34 1265-1269. · Zbl 0137.12704 · doi:10.1214/aoms/1177703862 |
| [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 Bay es estimators for a class of discrete distributions with monotone likelihood ratio. Statist. Neerlandica 31 95-104. · Zbl 0364.62030 · doi:10.1111/j.1467-9574.1977.tb00756.x |
| [102] | VAN Ry ZIN, J. R. (1966a). The sequential compound decision problem with m\times n finite loss matrix. Ann. Math. Statist. 37 954-975. · Zbl 0173.46303 · doi:10.1214/aoms/1177699376 |
| [103] | VAN Ry ZIN, J. R. (1966b). Repetitive play in finite statistical games with unknown distributions. Ann. Math. Statist. 37 976-994. · Zbl 0173.47802 · doi:10.1214/aoms/1177699377 |
| [104] | VAN Ry ZIN, J. R. and SUSARLA, V. (1977). On the empirical Bay es approach to multiple decision problems. Ann. Statist. 5 172-181. · Zbl 0379.62009 · doi:10.1214/aos/1176343750 |
| [105] | VARDEMAN, S. B. (1978). Admissible solutions of finite state sequence compound decision problems. Ann. Statist. 6 673-679. · Zbl 0388.62010 · doi:10.1214/aos/1176344211 |
| [106] | WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia. · Zbl 0813.62001 |
| [107] | WIND, S. L. (1973). An empirical Bay es approach to multiple linear regression. Ann. Statist. 1 93- 103. · Zbl 0253.62007 · doi:10.1214/aos/1193342385 |
| [108] | ZASLAVSKY, A. M. (1993). Combining census, dual-sy stem, 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. · Zbl 0778.62037 · doi:10.1214/aos/1176347627 |
| [110] | ZHANG, C.-H. (1995). On estimating mixing densities in discrete exponential family models. Ann. Statist. 23 929-945. · Zbl 0841.62027 · doi:10.1214/aos/1176324629 |
| [111] | ZHANG, C.-H. (1997). Empirical Bay es and compound estimation of normal means. Statist. Sinica 7 181-193. · Zbl 0904.62008 |
| [112] | ZHANG, C.-H. (2000). General empirical Bay es wavelet methods. Technical Report 2000-007, Dept. Statist., Rutgers Univ., Piscataway, NJ. |
| [113] | HILL CENTER, BUSCH CAMPUS PISCATAWAY, NEW JERSEY 08854 E-MAIL: czhang@stat.rutgers.edu |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
