Admissibility in quadratically regular problems and recurrence of symmetric Markov chains: Why the connection? (English) Zbl 0944.62010
Summary: In this expository paper, a sufficient condition is discussed for the almost admissibility of formal Bayes rules in quadratically regular problems. This sufficient condition is equivalent to a recurrence property of a natural symmetric Markov chain constructed from the model and the improper prior. Some simple examples involving translation parameter models illustrate the results.
MSC:
| 62C15 | Admissibility in statistical decision theory |
| 62C05 | General considerations in statistical decision theory |
| 62C10 | Bayesian problems; characterization of Bayes procedures |
| 62J10 | Analysis of variance and covariance (ANOVA) |
Keywords:
improper prior distributions; formal posterior distributions; recurrence; transience; formal Bayes rules; symmetric Markov chainReferences:
| [1] | Berger, J. O.; Lu, K. L., Estimated confidence procedures for multivariate normal means, J. Statist. Plann. Inference, 23, 1-19 (1989) · Zbl 0687.62022 |
| [2] | Berger, J.; Srinivasan, C., Generalized Bayes estimators in multivariate problems, Ann. Statist., 6, 783-801 (1978) · Zbl 0378.62004 |
| [3] | Blyth, C., On minimax statistical procedures and their admissibility, Ann. Math. Statist., 22, 22-42 (1951) · Zbl 0042.38303 |
| [4] | Correction. Correction, Ann. Math. Statist., 1, 594-596 (1979) · Zbl 0262.62009 |
| [5] | Brown, L., A heuristic method for determining admissibility of estimators — with applications, Ann. Statist., 5, 960-994 (1979) · Zbl 0414.62011 |
| [6] | Brown, L. D.; Hwang, J. T., A unified admissibility proof, (Gupta, S. S.; Berger, J. O., Proc. 3rd Purdue Symp. On Statistical Decision Theory III (1982), Academic Press: Academic Press New York) · Zbl 0585.62016 |
| [7] | Chung, K. L.; Fuchs, W. H.J., On the distribution of values of sums of random variables, Mem. Amer. Math. Soc., No. 6 (1951) · Zbl 0042.37502 |
| [8] | Cohen-Brandwein, A.; Strawderman, W., Stein estimation: the spherically symmetric case, Statist. Sci., 5, 356-369 (1990) · Zbl 0955.62611 |
| [9] | Diaconis, P.; Stroock, D., Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1, 36-61 (1991) · Zbl 0731.60061 |
| [10] | Eaton, M. L., A method for evaluating improper prior distributions, (Gupta, S. S.; Berger, J. O., Statistical Decision Theory and Related Topics III, vol. 1 (1982), Academic Press: Academic Press New York), 329-352 · Zbl 0581.62005 |
| [11] | Eaton, M. L., A statistical diptych: Admissible inferences — recurrence of symmetric Markov chains, Ann. Statist., 20, 1147-1179 (1992) · Zbl 0767.62002 |
| [12] | Eaton, M. L., On the inadmissibility of some best invariant predictive distributions. The translation parameter case, (Ghosh, J. K.; Mitra, S. K., Festschrift Volume for R.R. Bahadur (1993), Wiley Eastern Ltd), 191-202 |
| [13] | Feller, W., (An Introduction to Probability Theory and its Applications, vol II (1966), Wiley: Wiley New York) · Zbl 0138.10207 |
| [14] | Fourdringer, D.; Wells, M. T., Estimation of a loss function for spherically symmetric distributions on the general linear model, Ann. Statist., 23, 571-592 (1995) · Zbl 0829.62011 |
| [15] | Fukushima, M., Dirichlet Forms and Markov Processes (1980), North-Holland: North-Holland Amsterdam · Zbl 0422.31007 |
| [16] | Ghosh, M.; Meeden, G., Admissibility of linear estimators in the one parameter exponential family, Ann. Statist., 5, 772-778 (1977) · Zbl 0385.62006 |
| [17] | Guivarc’h, Y.; Keane, M.; Roynette, B., Marches aléatoires sur les Groupes de Lie, (Lecture Notes in Math., vol. 624 (1977), Springer: Springer New York) · Zbl 0367.60081 |
| [18] | Hwang, J. T.; Casella, G., Minimax confidence sets for the mean of a multivariate normal distribution, Ann. Statist., 10, 868-881 (1982) · Zbl 0508.62031 |
| [19] | Hwang, J. T.; Casella, G., Improved set estimators for a multivariate normal mean, Statistics and Decisions, Suppl. 1, 3-16 (1984) · Zbl 0559.62031 |
| [20] | Hwang, J. T.; Casella, G.; Robert, C.; Wells, M. T.; Farrell, R. H., Estimation of accuracy in testing, Ann. Statist., 20, 490-509 (1992) · Zbl 0761.62022 |
| [21] | Johnstone, I., Admissibility, difference equations, and recurrence in estimating a Poisson mean, Ann. Statist., 12, 1173-1198 (1984) · Zbl 0557.62006 |
| [22] | Johnstone, I., Admissible estimation, Dirichlet principles, and recurrence of birth-death chains on \(Z^p_+\), Probab. Theory Related Fields, 71, 231-269 (1986) · Zbl 0592.62009 |
| [23] | Johnstone, I., On inadmissibility of some unbiased estimates of loss, (Gupta, S. S.; Berger, J. O., Statistical Decision Theory and Related Topics IV, vol. 1 (1988), Springer: Springer New York), 361-379 · Zbl 0647.62018 |
| [24] | Karlin, S., Admissibility for estimation with quadratic loss, Ann. Math. Statist., 29, 406-436 (1958) · Zbl 0131.17805 |
| [25] | Kiefer, J., Conditional confidence statements and confidence estimators (with discussion), J. Amer. Statist. Assoc., 72, 789-827 (1977) · Zbl 0375.62023 |
| [26] | Kroese, A. H., Distributional inference: A loss function approach, (Ph.D. thesis (1994), Department of Mathematics, University of Groningen: Department of Mathematics, University of Groningen The Netherlands) |
| [27] | Mouchart, M., A note on Bayes theorem, Statistica, 36, 2, 349-357 (1976) · Zbl 0335.62003 |
| [28] | Srinivasan, C., Admissible generalized Bayes estimators and exterior boundary value problems, Sankhya Ser. A, 43, 1-25 (1981) · Zbl 0513.62009 |
| [29] | Zidek, J., Sufficient conditions for admissibility under squared error loss of formal Bayes estimators, Ann. Math. Statist., 41, 446-456 (1970) · Zbl 0196.21504 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
