×

zbMATH — the first resource for mathematics

Intrinsic posterior regret gamma-minimax estimation for the exponential family of distributions. (English) Zbl 1293.62048
Summary: In practice, it is desired to have estimates that are invariant under reparameterization. The invariance property of the estimators helps to formulate a unified solution to the underlying estimation problem. In robust Bayesian analysis, a frequent criticism is that the optimal estimators are not invariant under smooth reparameterizations. This paper considers the problem of posterior regret gamma-minimax (PRGM) estimation of the natural parameter of the exponential family of distributions under intrinsic loss functions with Kullback-Leibler distance. We show that under the class of Jeffrey’s Conjugate Prior (JCP) distributions, PRGM estimators are invariant to smooth one-to-one reparameterizations. We apply our results to several distributions and different classes of JCP, as well as the usual conjugate prior distributions. We observe that, in many cases, invariant PRGM estimators in the class of JCP distributions can be obtained by some modifications of PRGM estimators in the usual class of conjugate priors. Moreover, when the class of priors are convex or dependant on a hyper-parameter belonging to a connected set, we show that the PRGM estimator under the intrinsic loss function could be Bayes with respect to a prior distribution in the original prior class. Theoretical results are supplemented with several examples and illustrations.

MSC:
62F10 Point estimation
62F15 Bayesian inference
62C20 Minimax procedures in statistical decision theory
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Berger, J.O. (1994). An overview of robust Bayesian analysis . Test, 3 , 5-124. · Zbl 0827.62026
[2] Bernardo, J.M. (2011). Integrated objective Bayesian estimation and hypothesis testing (with discussion) . In Bayesian Analysis 9 (eds. J.M. Bernardo, M.J. Bayarri, J.O. Berger, A.P. Dawid, D. Heckerman, A.F.M. Smith and M. West). Oxford University Press, 1-68.
[3] Bernardo, J.M. and Smith, A.F.M. (1994). Bayesian Theory . Chichester: Wiley. · Zbl 0796.62002
[4] Boratyńska, A. (2002). Posterior regret gamma-minimax estimation in a normal model with asymmetric loss function. Acta Mathematicae, 29 , 7-13. · Zbl 1053.62015
[5] Boratyńska, A. (2006). Robust Bayesian prediction with asymmetric loss function in Poisson model of insurance risk . Acta Universitatis Lodziensis, Folia Oeconomica, 196 , 123-138.
[6] Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families . Annals of Statistics, 7 , 269-281. · Zbl 0405.62011
[7] Druilhet, P. and Pommeret, D. (2012). Invariant conjugate analysis for exponential families . Bayesian Analysis, 7 , 235-248. · Zbl 1330.62119
[8] Gelman, A. (2004). Parameterization and Bayesian modelling . Journal of the American Statistical Association, 99 , 537-545. · Zbl 1117.62343
[9] Gómez-Déniz, E. (2009). Some Bayesian credibility premiums obtained by using posterior regret gamma-minimax methodology. Bayesian Analysis, 4 , 223-242. · Zbl 1330.62124
[10] Gutiérrez-Peña, E. (1992). Expected Logarithmic Divergence for Exponential Families . In Bayesian Statistics 4 (J.M. Bernardo, J.O. Berger, A.P. Dawid y A.F.M. Smith, eds.) Oxford: University Press, 669-674.
[11] Gutiérrez-Peña, E. and Smith, A.F.M. (1995). Conjugate Parametrizations for Natural Exponential Families. Journal of the American Statistical Association, 90 , 1347-1356. · Zbl 0868.62029
[12] Jafari Jozani, M., and Parsian, A. (2008). Posterior regret \(\Gamma\)-minimax estimation and prediction based on \(k\)-record data under entropy loss function . Communications in Statistics: Theory and Methods, 37, 14 , 2202-2212. · Zbl 1143.62002
[13] Robert, C.P. (1996). Intrinsic loss functions . Theory and Decision, 40 , 192-214. · Zbl 0848.90010
[14] Ríos Insua, D., and Ruggeri, F. (2000). Robust Bayesian analysis . Lecture Notes in Statistics 152 , Springer-Verlag, New York. · Zbl 0958.00015
[15] Ríos Insua, D., Ruggeri, F., and Vidakovic, B. (1995). Some results on posterior regret \(\Gamma\)-minimax estimation . Statistics & Decisions, 13 , 315-351. · Zbl 0843.62008
[16] Zen, M., and DasGupta, A. (1993). Estimating a binomial parameter: Is robust Bayes real Bayes? Statistics & Decisions, 11 , 37-60. · Zbl 0767.62003
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.