Asymptotic Theory and the Foundations of Statistics

Reid, N.

Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 231-243

Résumé

Statistics in the 20th century has been enlivened by a passionate, occasionally bitter, and still vibrant debate on the foundations of statistics and in particular on Bayesian vs. frequentist approaches to inference. In 1975 D. V. Lindley predicted a Bayesian 21st century for statistics. This prediction has often been discussed since, but there is still no consensus on the probability of its correctness. Recent developments in the asymptotic theory of statistics are, surprisingly, shedding new light on this debate, and may have the potential to provide a common middle ground.

Détail

DOI : 10.4153/CMB-1997-028-x

Mots-clés : Primary: 62-02, Secondary: 62A10, 62A15

Bibliographie

[1] [1] Bickel, P. J. and Ghosh, J. K., A decomposition for the likelihood ratio statistic and Bartlett correction—a Bayesian argument, Ann. Statist. 18 (1990), 1070–1090. Google Scholar

[2] [2] Cox, D. R. and N. Reid, Parameter orthogonality and approximate conditional inference, J. R. Statist. Soc. B 49 (1987), 1–39. Google Scholar

[3] [3] DiCiccio, T. J. and Martin, M. A., Simple modifications for signed roots of likelihood ratio statistics, J. R. Statist. Soc. B 55 (1993), 305–316. Google Scholar

[4] [4] Efron, B., Controversies in the foundations of statistics, Amer. Math. Monthly 85 (1978), 231–246. Google Scholar

[5] [5] Efron, B., Why isn't everyone a Bayesian? Amer. Statist. 40 (1986), 1–11. Google Scholar

[6] [6] Efron, B., Bayes and likelihood calculations from confidence intervals, Biometrika 80 (1993a), 3–26. Google Scholar

[7] [7] Efron, B., Statistics in the 21st century, Statistics and Computing 3 (1993b), 188–190. Google Scholar

[8] [8] Efron, B., Empirical Bayes methods for combining likelihoods, J. Am. Statist. Assoc. 91 (1996), 538–565. Google Scholar

[9] [9] Fisher, R. A., On the ‘Probable Error’ of a coefficient of correlation deduced from a small sample, Metron I (1921), 3–32, Reprinted in Fisher (1950). Google Scholar

[10] [10] Fisher, R. A., On the mathematical foundations of theoretical statistics, Phil. Trans. Roy. Soc. A 222 (1922), 309–368, Reprinted in Fisher (1950). Google Scholar

[11] [11] Fisher, R. A., Theory of statistical estimation, Proc. Camb. Phil. Soc. 22 (1925), 700–725, Reprinted in Fisher (1950). Google Scholar

[12] [12] Fisher, R. A., Inverse probability, Proc. Camb. Phil. Soc. 26 (1930), 528–535, Reprinted in Fisher (1950). Google Scholar

[13] [13] Fisher, R. A., Contributions to Mathematical Statistics, J. Wiley & Sons, New York, 1950. Google Scholar

[14] [14] Ghosh, J. K., Higher order asymptotics, NSF-CBMSRegional Conference Series in Probability and Statistics, Volume 4, Institute of Mathematical Statistics, Hayward, 1994. Google Scholar

[15] [15] Jeffreys, H., Scientific Inference, 2nd edition 1957, Cambridge University Press, 1931. Google Scholar

[16] [16] Lane, D. A., Fisher, Jeffreys, and the nature of probability. In: R. A. Fisher, an appreciation, (eds. D. V. Hinkley and S. E. Fienberg), Springer-Verlag, New York, 1980. Google Scholar

[17] [17] Lindley, D. V., The future of statistics—a Bayesian 21st century. In: Proceedings of the Conference on Directions for Mathematical Statistics, (ed. S. G. Ghurye), Special Supplement to Adv. Appl. Probab., 1975. Google Scholar

[18] [18] Lindley, D. V., Contribution to the discussion of Efron (1986), Amer. Statist. 40 (1986), 6–7. Google Scholar

[19] [19] Peers, H. W., On confidence points and Bayesian probability points in the case of several parameters, J. R. Statist. Soc. B 27 (1965), 16–27. Google Scholar

[20] [20] Reid, N., Saddlepoint methods and statistical inference, Statist. Science 3 (1988), 213–238. Google Scholar

[21] [21] Reid, N., Likelihood and Bayesian approximation methods. In: Bayesian Statistics 351–368, (eds. J. O. Berger, J. M. Bernardo, D. V. Lindley and A. F. M. Smith), Oxford University Press, 1995. Google Scholar

[22] [22] Savage, L. J., The Foundations of Statistics, J. Wiley & Sons, New York, 1954. Google Scholar

[23] [23] Smith, A. F. M., Present position and potential developments: some personal views. Bayesian statistics, J. R. Statist. Soc. A (2)147 (1984), 245–259. Google Scholar

[24] [24] Stein, C., On the coverage probability of confidence sets based on a prior distribution. In: Sequential Methods in Statistics, Banach Center publications, 16, PWN-Polish Scientific Publishers, Warsaw, 1985. Google Scholar

[25] [25] Tibshirani, R. J., Non-informative priors for one parameter of many, Biometrika 76 (1989), 604–608. Google Scholar

[26] [26] Welch, B. L. and Peers, H. W., On formulae for confidence points based on integrals of weighted likelihoods, J. R. Statist. Soc. B 25 (1963), 318–329. Google Scholar