• Bernoulli
  • Volume 5, Number 5 (1999), 833-854.

Consistency and strong inconsistency of group-invariant predictive inferences

Morris L. Eaton and William D. Sudderth

Full-text: Open access


Consider a statistical model which is invariant under a group of transformations that acts transitively on the parameter space. In this situation, the problem of constructing invariant predictive distributions is considered. It is shown, under certain assumptions, that Fisherian pivoting and the use of right Haar measure as an improper prior distribution both yield the same invariant predictive distribution. Furthermore, it is shown that any other invariant predictive distribution is strongly inconsistent in the sense of Stone.

Article information

Bernoulli, Volume 5, Number 5 (1999), 833-854.

First available in Project Euclid: 12 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Fisherian pivoting improper prior distributions invariant predictive distribution proper action right Haar measure strong inconsistency


Eaton, Morris L.; Sudderth, William D. Consistency and strong inconsistency of group-invariant predictive inferences. Bernoulli 5 (1999), no. 5, 833--854.

Export citation


  • [1] Aitchison, J. and Dunsmore, I.R. (1975) Statistical Prediction Analysis. Cambridge: Cambridge University Press.
  • [2] Andersson, S. (1982) Distributions of maximal invariants using quotient measures. Ann. Statist., 10, 955-961.
  • [3] Andersson, S., Brons, H. and Jensen, S.T. (1983) Distribution of eigenvalues in multivariate statistical analysis. Ann. Statist., 11, 392-415.
  • [4] Barndorff-Nielsen, O.E. and Cox, D.R. (1996) Prediction and asymptotics. Bernoulli, 2, 319-340.
  • [5] Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd edition. New York: Springer-Verlag.
  • [6] Bondar, J.V. and Milnes, P. (1981) Amenability: A survey for statistical applications of Hunt-Stein and related conditions on groups. Z. Wahrscheinlichkeitstheorie Verw. Geb., 57, 103-128.
  • [7] Bourbaki, N. (1966) Elements of Mathematics. General Topology, Part 1. Reading, MA: Addison Wesley.
  • [8] Dunford, N. and Schwartz, J. (1957) Linear Operators, Part 1. New York: Interscience.
  • [9] Eaton, M.L. (1982) A method for evaluating improper prior distributions. In S.S. Gupta and J.O. Berger (eds), Statistical Decision Theory and Related Topics III, 1, pp. 329-352. New York: Academic Press.
  • [10] Eaton, M.L. (1983) Multivariate Statistics. New York: Wiley.
  • [11] Eaton, M.L. (1989) Group Invariance Applications in Statistics. Regional Conference Series in Probability and Statistics 1. Hayward, CA: Institute of Mathematical Statistics.
  • [12] Eaton, M.L. (1992) A statistical diptych: Admissible inferences - recurrence of symmetric Markov chains. Ann. Statist., 20, 1147-1179.
  • [13] Eaton, M.L. and Sudderth, W. (1993) Prediction in a multivariate normal setting: Coherence and incoherence. Sankhya Ser. A, Special Volume 55, Part 3, 481-493.
  • [14] Eaton, M.L. and Sudderth, W.D. (1995) The formal posterior of a standard flat prior in MANOVA is incoherent. J. Italian Statist. Soc., 2, 251-270.
  • [15] Fraser, D.A.S. (1968) The Structure of Inference. New York: Wiley.
  • [16] Geisser, S. (1993) Predictive Inference: An Introduction. New York: Chapman & Hall.
  • [17] Johnson, B.W. (1991) On the admissibility of improper Bayes inferences in fair Bayes decision problems. Ph.D. thesis, University of Minnesota.
  • [18] Kass, R. and Wasserman, L. (1996) The selection of prior distributions by formal rules. J. Amer. Statist. Assoc., 91, 1343-1370.
  • [19] Kiefer, J. (1975) Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. Statist., 28, 573-601.
  • [20] Lane, D. and Sudderth, W. (1984) Coherent predictive inference. Sankhya, Ser. A, 46, 166-185.
  • [21] Stein, C. (1965) Approximation of improper prior measures by prior probability measures. In L.M. LeCam and J. Neyman (eds), Bernoulli (1713) - Bayes (1763) - Laplace (1813), pp. 217-240. New York: Springer-Verlag.
  • [22] Stone, M. (1976) Strong inconsistency from uniform priors. J. Amer. Statist. Assoc., 71, 114-119.
  • [23] Wijsman, R.A. (1990) Invariant Measures on Groups and Their Use in Statistics. IMS Lecture Notes - Monograph Series 14. Hayward, CA: Institute of Mathematical Statistics.