## Robust Bayesian estimators in a one-way ANOVA model.(English)Zbl 0839.62024

Summary: Motivated by the attractive features of robust priors, we develop Bayesian estimators for the parameters in a one-way ANOVA model using mixed priors, which are formed by incorporating a $$t$$ density into the usual conjugate priors to independently describe prior knowledge regarding the overall mean or regarding the factor effects. The effect of the independent $$t$$ prior component is greatly different from that of the conjugate prior. The Bayesian estimators arising from such mixed priors are nonlinear functions of the least squares estimators and adjust automatically to the value of the sum of squared errors. In this sense, they are adaptive and rather insensitive to extreme observations. The proposed estimators are clearly superior to the usual Bayesian estimators and to the traditional unbiased estimators, and may be practicable when the error terms are Cauchy distributed.

### MSC:

 62F15 Bayesian inference 62J10 Analysis of variance and covariance (ANOVA) 62F35 Robustness and adaptive procedures (parametric inference)
Full Text:

### References:

 [1] Angers, J. F. and Berger, J. O. (1991). Robust hierarchical Bayesian estimation of exchangeable means.Canadian J. Statist. 19, 39–56. · Zbl 0719.62047 [2] Berger, J. O. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean.Ann. Statist. 8, 716–761. · Zbl 0464.62026 [3] Berger, J. O. (1984). The robust Bayesian viewpoint.Studies in Bayesian Econometrics 4 (A. Zellner and J. B. Kadane, eds) Amsterdam: North-Holland, 63–115. [4] Bian, G. (1989). Bayesian statistical analysis with independent bivariate priors for the normal location and scale parameters. Ph.D. Thesis, University of Minnesota. [5] Bian, G. and Dickey, J. M. (1995). Properties of multivariate Cauchy and poly-Cauchy distributions with Bayesian g-prior applications.Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner (D. A. Berry, K. M. Chaloner and J. K. Geweke, eds.), New York: Wiley. (to appear). [6] Broemeling, L. D. (1985)Bayesian Analysis of Linear Models. New York: Marcel Dekker. · Zbl 0564.62020 [7] Dickey, J. M. (1974). Bayesian alternatives to the F-test and least-squares estimate in the normal linear model.Studies in Bayesian Econometrics and Statistics (S. E. Fienberg and A. Zellner, eds.), Amsterdam: North-Holland, 515–554. [8] Dreze, J. H. and Richard, J. F. (1983). Bayesian analysis of simultaneous equation system.Handbook of Econometrics 1 (Z. Glriliches and M. D. Intriligator, eds.). Amsterdam: North-Holland, 577–598. [9] Helfand, A. E., Hills, S. E., Racine-poon, A., and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data model using Gibbs sampling.J. Amer. Statist. Assoc. 85, 972–985. [10] Jeffrey, H. (1948).Theory of Probability. 2nd edition. Oxford: University Press. [11] O’Hagan, A. (1990). Outliers and credence for location parameter inference.J. Amer. Statist. Assoc. 85, 172–176. · Zbl 0706.62030 [12] Patil, V. H. (1964). Difficulties involved in computing Behrens-Fisher densities, cumulative probabilities, and percentage points from first principles.J. Indian. Statist. Assoc. 2 and3, 110–118. [13] Press, S. J. (1989).Bayesian Statistics: Principles, Models, and Applications. New York: Wiley. · Zbl 0687.62001 [14] Raiffa, H., and Schlaifer, R. (1961).Applied Statistical Decision Theory. Boston: Harvard University. · Zbl 0952.62008 [15] Ramsay, J. O. and Novick, M. R. (1980). PLU robust Bayesian decision theory: point estimation.J. Amer. Statist. Assoc. 75, 401–407. · Zbl 0448.62003 [16] Rubin, H. (1977). Robust Bayesian estimation.Statistical Decision Theory and Related Topics II (S. S. Gupta and D. Moore, eds). New York: Academic Press, 351–356. [17] Tiao, G. C. and Zellner, A. (1964). On the Bayesian estimation of multivariate regression.J. Roy. Statist. Soc. B 26, 277–285. · Zbl 0136.39601 [18] Tirney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities.J. Amer. Statist. Assoc. 81, 82–86. · Zbl 0587.62067
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.