Eaton's Markov chain, its conjugate partner and $\mathscr{P}$-admissibility

James P. Hobert and C. P. Robert
Source: Ann. Statist. Volume 27, Number 1 (1999), 361-373.

Abstract

Suppose that X is a random variable with density $f(x|\theta)$ and that $\pi(\theta|x)$ is a proper posterior corresponding to an improper prior $\nu(\theta)$. The prior is called $\mathscr{P}$-admissible if the generalized Bayes estimator of every bounded function of $\theta$ is almost-$\nu$-admissible under squared error loss. Eaton showed that recurrence of the Markov chain with transition density $R(\eta|\theta) = \int \pi(\eta|x)f(x|\theta) dx$ is a sufficient condition for $\mathscr{P}$-admissibility of $\nu(\theta)$. We show that Eaton’s Markov chain is recurrent if and only if its conjugate partner, with transition density $\tilde{R}(y|x) = \int f(y|\theta) \pi(\theta|x) d\theta$, is recurrent. This provides a new method of establishing $\mathscr{P}$-admissibility. Often, one of these two Markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience. For example, when $X$ is Poisson $(\theta)$ and an improper gamma prior is placed on $\theta$, the Markov chain defined by $\tilde{R}(y|x)$ is equivalent to a branching process with immigration. We use this type of argument to establish $\mathscr{P}$-admissibility of some priors when $f$ is a negative binomial mass function and when $f$ is a gamma density with known shape.

First Page:
Primary Subjects: 62C15
Secondary Subjects: 60J05
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aos/1018031115
Mathematical Reviews number (MathSciNet): MR1701115
Digital Object Identifier: doi:10.1214/aos/1018031115
Zentralblatt MATH identifier: 0945.62012

References

Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation Xn = AnXn-1+ Bn in the critical case. Ann. Probab. 25 478-493.
Mathematical Reviews (MathSciNet): MR1428518
Zentralblatt MATH: 0873.60045
Digital Object Identifier: doi:10.1214/aop/1024404297
Project Euclid: euclid.aop/1024404297
Berger, J. O. (1980). Improving on inadmissible estimators in continuous exponential families with applications to simultaneous estimation of gamma scale parameters. Ann. Statist. 8 545-571.
Mathematical Reviews (MathSciNet): MR82d:62018
Zentralblatt MATH: 0447.62008
Digital Object Identifier: doi:10.1214/aos/1176345008
Project Euclid: euclid.aos/1176345008
Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. Springer, New York.
Mathematical Reviews (MathSciNet): MR87i:62003
Zentralblatt MATH: 0572.62008
Brandt, A. (1986). The stochastic equation Yn+1 = AnYn + Bn with stationary coefficients. Adv. Appl. Probab. 18 211-220.
Mathematical Reviews (MathSciNet): MR827336
Zentralblatt MATH: 0588.60056
Digital Object Identifier: doi:10.2307/1427243
Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855-904.
Mathematical Reviews (MathSciNet): MR44:3423
Zentralblatt MATH: 0246.62016
Digital Object Identifier: doi:10.1214/aoms/1177693318
Project Euclid: euclid.aoms/1177693318
Das Gupta, A. (1984). Admissibility in the gamma distribution: two examples. Sankhy¯a Ser. A 46 395-407
Mathematical Reviews (MathSciNet): MR86j:62021
Diebolt, J. and Robert, C. P. (1994). Estimation of finite mixture distributions by Bayesian sampling. J. Roy. Statist. Soc. Ser. B 56 363-375.
Mathematical Reviews (MathSciNet): MR1281940
Eaton, M. L. (1982). A Method for evaluating improper prior distributions. In Statistical Decision Theory and Related Topics III 1 (S. S. Gupta and J. O. Berger, eds.). Academic, New York, 329-352.
Mathematical Reviews (MathSciNet): MR84m:62012
Zentralblatt MATH: 0581.62005
Eaton, M. L. (1992). A statistical diptych: admissible inferences-recurrence of symmetric Markov chains. Ann. Statist. 20 1147-1179.
Mathematical Reviews (MathSciNet): MR93i:62006
Zentralblatt MATH: 0767.62002
Digital Object Identifier: doi:10.1214/aos/1176348764
Project Euclid: euclid.aos/1176348764
Eaton, M. L. (1997). Admissibility in quadratically regular problems and recurrence of symmetric Markov chains: Why the connection? J. Statist. Plann. Inference 64 231-247.
Mathematical Reviews (MathSciNet): MR1621615
Zentralblatt MATH: 0944.62010
Digital Object Identifier: doi:10.1016/S0378-3758(97)00037-2
Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
Mathematical Reviews (MathSciNet): MR228020
Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
Hwang, J. T. (1982). Improving upon standard estimators in discrete exponential families with applications to Poisson and negative binomial cases. Ann. Statist. 10 857-867.
Mathematical Reviews (MathSciNet): MR83k:62012
Zentralblatt MATH: 0493.62008
Digital Object Identifier: doi:10.1214/aos/1176345876
Project Euclid: euclid.aos/1176345876
Hwang, J. T. (1982). Semitail upper bounds on the class of admissible estimators in discrete exponential families with applications to Poisson and negative binomial distributions. Ann. Statist. 10 1137-1147.
Mathematical Reviews (MathSciNet): MR84j:62013
Digital Object Identifier: doi:10.1214/aos/1176345979
Project Euclid: euclid.aos/1176345979
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd ed. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1224449
Johnstone, I. (1984). Admissibility, difference equations, and recurrence in estimating a Poisson mean. Ann. Statist. 12 1173-1198.
Mathematical Reviews (MathSciNet): MR86e:62013
Zentralblatt MATH: 0557.62006
Digital Object Identifier: doi:10.1214/aos/1176346786
Project Euclid: euclid.aos/1176346786
Johnstone, I. (1986). Admissible estimation, dirichlet principles, and recurrence of birth-death chains on Zpt. Probab. Theory Related Fields 71 231-269.
Mathematical Reviews (MathSciNet): MR816705
Zentralblatt MATH: 0592.62009
Digital Object Identifier: doi:10.1007/BF00332311
Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.
Mathematical Reviews (MathSciNet): MR81j:60105
Zentralblatt MATH: 0422.60001
Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Probab. 23 614- 625.
Mathematical Reviews (MathSciNet): MR88h:60086
Zentralblatt MATH: 0611.60084
Digital Object Identifier: doi:10.2307/3214001
Lai, W.-L. (1996). Admissibility and the recurrence of Markov chains with applications. Technical Report 612, School of Statistics, Univ. Minnesota.
Lamperti, J. (1960). Criteria for the recurrence or transience of stochastic processes I. J. Math. Anal. Appl. 1 314-330.
Mathematical Reviews (MathSciNet): MR23:A4166
Zentralblatt MATH: 0099.12901
Digital Object Identifier: doi:10.1016/0022-247X(60)90005-6
Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet): MR1639875
Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81 27-40.
Mathematical Reviews (MathSciNet): MR95d:62133
Zentralblatt MATH: 0811.62080
Digital Object Identifier: doi:10.1093/biomet/81.1.27
Lyons, T. (1983). A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11 393-402.
Mathematical Reviews (MathSciNet): MR84e:60102
Zentralblatt MATH: 0509.60067
Digital Object Identifier: doi:10.1214/aop/1176993604
Project Euclid: euclid.aop/1176993604
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
Mathematical Reviews (MathSciNet): MR95j:60103
Pakes, A. G. (1971). On the critical Galton-Watson process with immigration. J. Austral. Math. Soc. 12 476-482.
Mathematical Reviews (MathSciNet): MR46:6490
Digital Object Identifier: doi:10.1017/S1446788700010375
Panaretos, J. and Xekalaki, E. (1986). On some distributions arising from certain generalized sampling schemes. Comm. Statist. Theory Methods 15 873-891.
Mathematical Reviews (MathSciNet): MR87d:62029
Zentralblatt MATH: 0612.60014
Digital Object Identifier: doi:10.1080/03610928608829157
Stein, C. (1965). Approximation of improper prior measures by prior probability measures. In Bernoulli-Bayes-Laplace Festschrift (J. Neyman and L. Le Cam, eds.) 217-240. Springer, New York.
Mathematical Reviews (MathSciNet): MR33:8077
Zentralblatt MATH: 0139.36004
Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc. 52 528-550.
Mathematical Reviews (MathSciNet): MR898357
Zentralblatt MATH: 0619.62029
Digital Object Identifier: doi:10.2307/2289457
CREST, INSEE 75675 Paris cedex 14 France E-mail: robert@ensae.fr