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Bayes method
In this paper empirical Bayes methods are applied to construct selection rules for the selection of all good exponential distributions. We modify the selection rule introduced and studied by Gupta and Liang [10] who proved that the regret risk converges to zero with rate $O(n^{-\lambda /2}),0<\lambda \le 2$. The aim of this paper is to study the asymptotic behavior of the conditional regret risk ${\cal R}_{n}$. It is shown that $n{\cal R}_{n}$ tends in distribution to a linear combination of independent $\chi ^{2}$-distributed random variables. As an application we give a large sample approximation for the probability that the conditional regret risk exceeds the Bayes risk by a given $\varepsilon >0.$ This probability characterizes the information contained in the historical data.
[1] Balakrishnan N., (eds.) A. P. Basu: The Exponential Distribution: Theory, Method and Applications. Gordon and Breach Publishers. Langliorne, Pennsylvania 1995 MR 1655093
[2] Deely J. J.: Multiple Decision Procedures from Empirical Bayes Approach. Ph.D. Thesis (Mimeo. Ser. No. 45). Dept. Statist., Purdue Univ., West Lafayette, Ind. 1965 MR 2615366
[3] Dvoretzky A., Kiefer, J., Wolfowitz J.: Asymptotic minimax character of the sample distribution functions and of the classical multinomial estimator. Ann. Math. Statist. 27 (1956), 642–669 DOI 10.1214/aoms/1177728174 | MR 0083864
[4] Gupta S. S., Panchapakesan S.: Subset selection procedures: review and assessment. Amer. J. Management Math. Sci. 5 (1985), 235–311 MR 0859941 | Zbl 0633.62024
[5] Gupta S. S., Liang T.: Empirical Bayes rules for selecting the best binomial population. In: Statistical Decision Theory and Related Topics IV (S. S. Gupta and J. O. Berger, eds.), Vol. 1. Springer–Verlag, Berlin 1986, pp. 213–224 MR 0927102
[6] Gupta S. S., Liang T.: On empirical Bayes selection rules for sampling inspection. J. Statist. Plann. Inference 38 (1994), 43–64 DOI 10.1016/0378-3758(92)00154-V | MR 1256847 | Zbl 0797.62004
[7] Gupta S. S., Liang, T., Rau R.-B.: Empirical Bayes two stage procedures for selecting the best Bernoulli population compared with a control. In: Statistical Decision Theory and Related Topics V. (S. S. Gupta and J. O. Berger, eds.), Springer–Verlag, Berlin 1994, pp. 277–292 MR 1286308 | Zbl 0788.62010
[8] Gupta S. S., Liang, T., Rau R.-B.: Empirical Bayes rules for selecting the best normal population compared with a control. Statist. Decision 12 (1994), 125–147 MR 1292660 | Zbl 0804.62009
[10] Gupta S. S., Liang T.: Selecting good exponential populations compared with a control: nonparametric empirical Bayes approach. Sankhya, Ser. B 61 (1999), 289–304 MR 1734172
[11] Ibragimov I. A., Has’minskii R. Z.: Statistical Estimation: Asymptotic Theory. Springer, New York 1981 MR 0620321
[12] Johnson N. L., Kotz S., Balakrishnan N.: Continuous Univariate Distributions, Vol. 1. Second edition. Wiley, New York 1994 MR 1299979 | Zbl 0821.62001
[13] Jurečková J., Sen P. K.: Robust Statistical Procedures, Asymptotics and Interrelations. Wiley, New York 1996 MR 1387346 | Zbl 0862.62032
[14] Liese F., Vajda I.: Consistency of M-estimates in general regression models. J. Multivariate Anal. 50 (1994), 93–114 DOI 10.1006/jmva.1994.1036 | MR 1292610 | Zbl 0872.62071
[15] Pfanzagl J.: On measurability and consistency of minimum contrast estimators. Metrika 14 (1969), 249–272 DOI 10.1007/BF02613654
[16] Pollard D.: Asymptotics for least absolute deviation regression estimators. Econometric Theory 7 (1991), 186–199 DOI 10.1017/S0266466600004394 | MR 1128411
[17] Robbins H.: An empirical Bayes approach to statistics. In: Proc. Third Berkeley Symp., Math. Statist. Probab. 1, Univ. of California Press 1956, pp. 157–163 MR 0084919 | Zbl 0074.35302
[18] Shorack G. R., Wellner J. A.: Empirical Processes with Applications to Statistics. Wiley, New York 1986 MR 0838963 | Zbl 1171.62057
[19] Wald A.: Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 (1949), 595–601 DOI 10.1214/aoms/1177729952 | MR 0032169 | Zbl 0034.22902
[20] Vaart A. W. van der, Wellner J. A.: Weak Convergence and Empirical Processes (Springer Series in Statistics). Springer–Verlag, Berlin 1996 MR 1385671
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