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Asymptotic distribution of the conditional regret risk for selecting good exponential populations
Kybernetika, Tome 36 (2000) no. 5, pp. 571-588 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 62C10, 62C12, 62E20, 62F07
Keywords: Bayes method 
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Gupta, Shanti S.; Liese, Friedrich. Asymptotic distribution of the conditional regret risk for selecting good exponential populations. Kybernetika, Tome 36 (2000) no. 5, pp. 571-588. http://geodesic.mathdoc.fr/item/KYB_2000_36_5_a4/

[1] Balakrishnan N., (eds.) A. P. Basu: The Exponential Distribution: Theory, Method and Applications. Gordon and Breach Publishers. Langliorne, Pennsylvania 1995 | MR

[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

[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 | MR

[4] Gupta S. S., Panchapakesan S.: Subset selection procedures: review and assessment. Amer. J. Management Math. Sci. 5 (1985), 235–311 | MR | Zbl

[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

[6] Gupta S. S., Liang T.: On empirical Bayes selection rules for sampling inspection. J. Statist. Plann. Inference 38 (1994), 43–64 | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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

[11] Ibragimov I. A., Has’minskii R. Z.: Statistical Estimation: Asymptotic Theory. Springer, New York 1981 | MR

[12] Johnson N. L., Kotz S., Balakrishnan N.: Continuous Univariate Distributions, Vol. 1. Second edition. Wiley, New York 1994 | MR | Zbl

[13] Jurečková J., Sen P. K.: Robust Statistical Procedures, Asymptotics and Interrelations. Wiley, New York 1996 | MR | Zbl

[14] Liese F., Vajda I.: Consistency of M-estimates in general regression models. J. Multivariate Anal. 50 (1994), 93–114 | DOI | MR | Zbl

[15] Pfanzagl J.: On measurability and consistency of minimum contrast estimators. Metrika 14 (1969), 249–272 | DOI

[16] Pollard D.: Asymptotics for least absolute deviation regression estimators. Econometric Theory 7 (1991), 186–199 | DOI | MR

[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 | Zbl

[18] Shorack G. R., Wellner J. A.: Empirical Processes with Applications to Statistics. Wiley, New York 1986 | MR | Zbl

[19] Wald A.: Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 (1949), 595–601 | DOI | MR | Zbl

[20] Vaart A. W. van der, Wellner J. A.: Weak Convergence and Empirical Processes (Springer Series in Statistics). Springer–Verlag, Berlin 1996 | MR