## Simultaneous prediction of independent Poisson observables.(English)Zbl 1092.62036

Summary: Simultaneous predictive distributions for independent Poisson observables are investigated. A class of improper prior distributions for Poisson means is introduced. The Bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the Kullback-Leibler loss. A Bayesian predictive distribution based on a prior in this class dominates the Bayesian predictive distribution based on the Jeffreys prior.

### MSC:

 62F15 Bayesian inference 62C15 Admissibility in statistical decision theory
Full Text:

### References:

 [1] Aitchison, J. (1975). Goodness of prediction fit. Biometrika 62 547–554. · Zbl 0339.62018 [2] Aitchison, J. and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge Univ. Press. · Zbl 0327.62043 [3] Akaike, H. (1978). A new look at the Bayes procedure. Biometrika 65 53–59. · Zbl 0373.62008 [4] Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 200–207. · Zbl 0682.62018 [5] Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 41 113–147. · Zbl 0428.62004 [6] Blyth, C. R. (1951). On minimax statistical decision procedures and their admissibility. Ann. Math. Statist. 22 22–42. · Zbl 0042.38303 [7] Brown, L. D. and Hwang, J. T. (1982). A unified admissibility proof. In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, eds.) 1 205–230. Academic Press, New York. · Zbl 0585.62016 [8] Clarke, B. S. and Barron, A. R. (1994). Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Statist. Plann. Inference 41 37–60. · Zbl 0820.62006 [9] Clevenson, M. L. and Zidek, J. V. (1975). Simultaneous estimation of the means of independent Poisson laws. J. Amer. Statist. Assoc. 70 698–705. · Zbl 0308.62018 [10] Dawid, A. P. (1984). Statistical theory. The prequential approach (with discussion). J. Roy. Statist. Soc. Ser. A 147 278–292. · Zbl 0557.62080 [11] Geisser, S. (1993). Predictive Inference : An Introduction. Chapman and Hall, New York. · Zbl 0824.62001 [12] Ghosh, M. and Yang, M.-C. (1988). Simultaneous estimation of Poisson means under entropy loss. Ann. Statist. 16 278–291. JSTOR: · Zbl 0654.62012 [13] Hartigan, J. A. (1965). The asymptotically unbiased prior distribution. Ann. Math. Statist. 36 1137–1152. · Zbl 0133.42106 [14] Hartigan, J. A. (1998). The maximum likelihood prior. Ann. Statist. 26 2083–2103. · Zbl 0927.62023 [15] Haussler, D. and Opper, M. (1997). Mutual information, metric entropy and cumulative relative entropy risk. Ann. Statist. 25 2451–2492. · Zbl 0920.62007 [16] Ibragimov, I. A. and Hasminskii, R. Z. (1973). On the information contained in a sample about a parameter. In Second International Symposium on Information Theory (B. N. Petrov and F. Csáki, eds.) 295–309. Akadémiai Kiado, Budapest. · Zbl 0289.62010 [17] James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361–380. Univ. California Press, Berkeley. · Zbl 1281.62026 [18] Jeffreys, H. (1961). Theory of Probability , 3rd ed. Oxford Univ. Press. · Zbl 0116.34904 [19] Komaki, F. (1996). On asymptotic properties of predictive distributions. Biometrika 83 299–313. · Zbl 0864.62007 [20] Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables. Biometrika 88 859–864. · Zbl 0985.62024 [21] Komaki, F. (2002a). Bayesian predictive distribution with right invariant priors. Calcutta Statist. Assoc. Bull. 52 171–179. [22] Komaki, F. (2002b). Shrinkage priors for Bayesian prediction. Unpublished manuscript. · Zbl 1092.62037 [23] Murray, G. D. (1977). A note on the estimation of probability density functions. Biometrika 64 150–152. · Zbl 0347.62035 [24] Ng, V. M. (1980). On the estimation of parametric density functions. Biometrika 67 505–506. · Zbl 0451.62006 [25] Skouras, K. and Dawid, A. P. (1999). On efficient probability forecasting systems. Biometrika 86 765–784. · Zbl 0948.62067 [26] Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate distribution. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197–206. Univ. California Press, Berkeley. · Zbl 0073.35602 [27] Stein, C. (1974). Estimation of the mean of a multivariate normal distribution. In Proc. Prague Symposium on Asymptotic Statistics (J. Hájek, ed.) 2 345–381. Univ. Karlova, Prague. · Zbl 0357.62020 [28] Vidoni, P. (1995). A simple predictive density based on the $$p^*$$-formula. Biometrika 82 855–863. · Zbl 0878.62017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.