Theory and Decision

, Volume 40, Issue 2, pp 191–214 | Cite as

Intrinsic losses

  • Christian P. Robert
Article

Abstract

Since the choice of a particular loss function strongly influences the resulting inference, it seems necessary to rely on “intrinsic” losses when no information is available about the utility function of the decision-maker, rather than to call for classical losses like the squared error loss. Since this setting is quite similar to the derivation of noninformative priors in Bayesian analysis, we first recall the conditions of this derivation and deduce from these conditions some requirements on the intrinsic losses. It then appears that these loss functions should only depend on the sampling distribution and that they should be independent of the parameterization of the distribution. The resulting estimators are therefore transformation equivariant. We study the properties of two natural intrinsic losses, namely entropy and Hellinger losses, and show that they can be expressed in closed form for exponential families. Moreover, the entropy loss also provides analytic expressions of Bayes estimators under conjugate priors; the derivation of Bayes estimators associated with the Hellinger loss is more cumbersome, as shown in Poisson and Gamma cases, while leading to similar estimators.

AMS Subject Classification (1990)

62J99 62F15 62C10 62C15 

Key words

Utility theory non-informative prior distributional distance entropy Hellinger distance conjugate prior Fisher information exponential families bayes estimator 

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References

  1. Abramowitz, M. and Stegun, I.: 1964, Handbook of Mathematical Functions, Dover.Google Scholar
  2. Berger, J.O.: 1985, Statistical Decision Theory and Bayesian Analysis (2nd edition). Springer-Verlag, New York.Google Scholar
  3. Berger, J.O. and Bernardo, J.M.: 1989, ‘Estimating a product of means Bayesian analysis with reference priors’, J. Amer. Statist. Assoc. 84, 200–207.Google Scholar
  4. Berger, J.O. and Wolpert, R.: 1988, The Likelihood Principle (2nd edition). IMS Lecture Notes, Monograph Series 6, Hay ward, California.Google Scholar
  5. Bernardo, J.M. 1979, ‘Reference posterior distributions for Bayesian inference (with discussion)’, J. Royal Statist. Soc. (Ser. B) 41, 113–147.Google Scholar
  6. Bernardo, J.M. and Smith, A.F.M.: 1994, Bayesian Theory, J. Wiley, New York.Google Scholar
  7. Birgé, L.: 1980, Approximation dans les espaces métriques et théorie de l'estimation, Thèse d'Etat, Univ. de Paris VIIGoogle Scholar
  8. Birgé, L.: 1983, ‘Robust testing for independent non identically distributed variables and Markov chains’, Lecture Notes in Statistics 16, Springer-Verlag, New York.Google Scholar
  9. Brown, L.D.: 1986, Foundations of Exponential Families, IMS Lecture Notes, Monograph Series 9, Hay ward, California.Google Scholar
  10. Brown, L.D.: 1980, ‘Examples of Berger's phenomenon in the estimation of independent normal means’, Ann. Statist. 9, 1289–1300.Google Scholar
  11. DeGroot, M.: 1970, Optimal Statistical Decisions, McGraw-Hill, New York.Google Scholar
  12. Fishburn, P.: 1988, Non-linear Preferences and Utility Theory, Harvester Wheatsheaf, Brighton, Sussex.Google Scholar
  13. Gauβ, C.F.: 1810, Méthode des moindres carrés. Mémoire sur la combination des observations, Trad. J. Bertrand. Mallet-Bachelier, Paris (1985).Google Scholar
  14. Gibbs, J.W.: 1876, Collected Works and Commentary, vol. II, A. Haas (Ed.) Yale University Press, New Haven (1936).Google Scholar
  15. Gouriéroux, C. and Monfort, A.: 1994, ‘Testing non-nested hypotheses’, Handbook of Econometrics, Vol. IV, R.F. Engle and D.L. McFadden (Eds.), Elsevier, Amsterdam.Google Scholar
  16. Gutiérrez-Peña, E.: 1992, ‘Expected logarithmic divergence for exponential families’, in Bayesian Statistics 4, J.O. Bernardo, A.P. Dawid and A.F.M. Smith (eds.). Oxford University Press, London, pp. 669–674.Google Scholar
  17. Huber, P.: 1964, ‘Robust estimation of a location parameter’, Ann. Math. Statist. 35, 73–101.Google Scholar
  18. Jaynes, E.t.: 1989, Papers on Probability, Statistics and Statistical Physics, R.D. Rosenkrantz (Ed.), Kluwer Academic Publishers, Dordrecht.Google Scholar
  19. Jeffreys, H.: 1961, Theory of Probability (3rd edition), Oxford University Press, London.Google Scholar
  20. Keating, J.P., Mason, R.L. and Sen, P.K.: 1993, Pitman Measure of Closeness: Comparison of Statistical Estimators, SIAM, Philadelphia.Google Scholar
  21. Le Cam, L.: 1982, ‘On the risk of Bayes estimates’, in Statistical Decision Theory and Related Topics III, Vol. 2, J.L. Berger and S.S. Gupta (Eds.) Academic Press, New York.Google Scholar
  22. Le Cam, L.: 1986, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York.Google Scholar
  23. Le Cam, L. and Yang, G.L.: 1990, Asymptotics in Statistics, Springer-Verlag, New York.Google Scholar
  24. Lehmann, E. Lehmann, E.L.: 1993, Theory of Point Estimation J. Wiley, New York.Google Scholar
  25. Lehmann, E.L. and Casella, G.: 1995, Theory of Point Estimation (revised edition). Wadsworth, Pacific Grove, California.Google Scholar
  26. Lindley, D.: 1985, Making Decisions, Wiley, New York.Google Scholar
  27. McCulloch, R. and Rossi, P.E.: 1992, ‘Bayes factors for nonlinear hypotheses and likelihood distributions’, Biometrika 79, 663–676.Google Scholar
  28. Olver, F.W.J.: 1974, Asymptotics and Special Functions, Academic-Press, New York.Google Scholar
  29. Pitman, E.: 1937, ‘The closest estimates of statistical parameters’, Proc. Cambridge Phil. Soc. 33, 212–222.Google Scholar
  30. Rao, C.R.: 1981, ‘Some comments on the minimum mean square error as a criterion of estimation’, in Statistics and Related Topics, M. Csörgo, D. Dawson, J.N.K. Rao, and A. Saleh (Eds.), pp. 123–143.Google Scholar
  31. Robert, C.P.: 1994, The Bayesian Choice, Springer-Verlag, New York.Google Scholar
  32. Robert, C.P., Hwang, J.T.G. and Strawderman, W.E.: 1993, ‘Is Pitman nearness a reasonable criterion? (with discussion)’, J. American Statistical Assoc. 88(1), 57–76.Google Scholar
  33. Rubin, H.: 1987, ‘A weak system of axioms for “rational” behavior and the nonseparability of utility from prior’, Statist. Decisions 5, 47–58.Google Scholar
  34. Rukhin, A.: 1978, ‘Universal Bayes estimators’, Ann. Statist. 6, 345–351.Google Scholar
  35. Shannon, C.: 1948, ‘A mathematical theory of communication’, Bell System Tech. J. 27, 379–423 and 623–656.Google Scholar
  36. Stigler, S.: 1986, The History of Statistics Belknap, Harvard.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Christian P. Robert
    • 1
  1. 1.INSEE, Centre de Recherche en Economie et StatistiqueMalakoff cedexFrance

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