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Asymptotic Normality with Small Relative Errors of Posterior Probabilities of Half-Spaces

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Selected Works of R.M. Dudley

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Abstract

Some examples of half-spaces of interest in parameter spaces are, in a clinical trial of a treatment versus placebo, the half-spaces where the treatment is (a) helpful or (b) harmful. Thus one may not only want to test the hypothesis that the treatment (c) makes no difference, but to assign posterior probabilities to (a), (b) and (c), under conditions as unrestrictive as possible on the choice of prior probabilities [e.g., Dudley and Haughton (2001)]. More generally, we have in mind applications to model selections as in the BIC criterion of Schwarz (1978) and its extensions [Poskitt (1987); Haughton (1988)], specifically to one-sided models and multiple data sets [Dudley and Haughton (1997)].

Received December 1999; revised October 2001.

Supported in part by NSF Grants DMS-97-04603 and DMS-01-03821.

Supported in part by an NSF grant.

AMS 2000 subject classifications. Primary 62F15; secondary 60F99, 62F05.

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References

  • Alon, N., Ben-David, S., Cesa-Bianchi, N. and Haussler, D. (1997). Scale-sensitive dimensions, uniform convergence, and learnability. J. ACM 44 615–631.

    Article  MATH  MathSciNet  Google Scholar 

  • Barndorff-Nielsen, O. E. and Wood, A. T. A. (1998). On large deviations and choice of ancillary for p * and r *. Bernoulli 4 35–63.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. and Mortera, J. (1999). Default Bayes factors for nonnested hypothesis testing. J. Amer. Statist. Assoc. 94 542–554.

    Article  MATH  MathSciNet  Google Scholar 

  • Berk, R. H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37 51–58. [Correction (1996) 37 745–746.]

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Bleistein, N. (1966). Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19 353–370.

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, C.-F. (1985). On asymptotic normality of limiting density functions with Bayesian implications. J. Roy. Statist. Soc. Ser. B 47 540–546.

    MATH  MathSciNet  Google Scholar 

  • Choy, S. T. B. and Smith, A. F. M. (1997). On robust analysis of a normal location parameter. J. Roy. Statist. Soc. Ser. B 59 463–474.

    Article  MATH  MathSciNet  Google Scholar 

  • Daniels, H. E. (1987). Tail probability approximations. Internat. Statist. Rev. 55 37–48.

    Article  MATH  MathSciNet  Google Scholar 

  • DiCiccio, T. J. and Martin, M. A. (1991). Approximations of marginal tail probabilities for a class of smooth functions with applications to Bayesian and conditional inference. Biometrika 78 891–902.

    Article  MATH  MathSciNet  Google Scholar 

  • DiCiccio, T. J. and Stern, S. E. (1993). On Bartlett adjustments for approximate Bayesian inference. Biometrika 80 731–740.

    Article  MATH  MathSciNet  Google Scholar 

  • Dudley, R. M. (1993). Real Analysis and Probability, 2nd ed., corrected. Chapman and Hall, New York.

    Google Scholar 

  • Dudley, R. M. (1998). Consistency of M-estimators and one-sided bracketing. In High Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.) 33–58. Birkhäuser, Basel.

    Google Scholar 

  • Dudley, R. M., Giné, E. and Zinn, J. (1991). Uniform and universal Glivenko–Cantelli classes. J. Theoret. Probab. 4 485–510.

    Article  MATH  MathSciNet  Google Scholar 

  • Dudley, R. M. and Haughton, D. (1997). Information criteria for multiple data sets and restricted parameters. Statist. Sinica 7 265–284.

    MATH  MathSciNet  Google Scholar 

  • Dudley, R. M. and Haughton, D. (2001). One-sided hypotheses in a multinomial model. In Goodness-of-Fit Tests and Model Validity (C. Huber-Carol, N. Balakrishnan, M. S. Nikulin and M. Mesbah, eds.) 387–399. Birkhäuser, Boston.

    Google Scholar 

  • Erkanli, A. (1994). Laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space. J. Amer. Statist. Assoc. 89 250–258.

    Article  MATH  MathSciNet  Google Scholar 

  • Fraser, D. A. S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86 249–264.

    Article  MATH  MathSciNet  Google Scholar 

  • Fulks, W. and Sather, J. O. (1961). Asymptotics II Laplace’s method for multiple integrals. Pacific J. Math. 11 185–192.

    MATH  MathSciNet  Google Scholar 

  • Haughton, D. (1984). On the choice of a model to fit data from an exponential family. Ph.D. dissertation, MIT.

    Google Scholar 

  • Haughton, D. M. A. (1988). On the choice of a model to fit data from an exponential family. Ann. Statist. 16 342–355.

    Article  MATH  MathSciNet  Google Scholar 

  • Hipp C. and Michel, R. (1976). On the Bernstein–v. Mises approximation of posterior distributions. Ann. Statist. 4 972–980.

    Article  MATH  MathSciNet  Google Scholar 

  • Holt, R. J. (1986). Computation of gamma and beta tail probabilities. Technical report, Dept. Mathematics, MIT.

    Google Scholar 

  • Hsu, L. C. (1948). A theorem on the asymptotic behavior of a multiple integral. Duke Math. J. 15 623–632.

    Article  MATH  MathSciNet  Google Scholar 

  • Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 221–233. Univ. California Press, Berkeley.

    Google Scholar 

  • Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press.

    Google Scholar 

  • Johnson, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Statist. 41 851–864.

    Article  MATH  MathSciNet  Google Scholar 

  • Lawley, D. N. (1956). A general method for approximating to the distribution of likelihood ratio criteria. Biometrika 43 295–303.

    MATH  MathSciNet  Google Scholar 

  • Le Cam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates. Univ. California Publ. Statist. 1 227–329.

    MathSciNet  Google Scholar 

  • Leonard, T. (1982). Comment. J. Amer. Statist. Assoc., 77 657–658.

    Article  MathSciNet  Google Scholar 

  • Lugannani, R. and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475–490.

    Article  MATH  MathSciNet  Google Scholar 

  • Pauler, D. K., Wakefield, J. C. and Kass, R. E. (1999). Bayes factors and approximations for variance component models. J. Amer. Statist. Assoc. 94 1242–1253.

    Article  MATH  MathSciNet  Google Scholar 

  • Pericchi, L. R. and Smith, A. F. M. (1992). Exact and approximate posterior moments for a normal location parameter. J. Roy. Statist. Soc. Ser. B. 54 793–804.

    MATH  MathSciNet  Google Scholar 

  • Poskitt, D. S. (1987). Precision, complexity and Bayesian model determination. J. Roy. Statis.t Soc. Ser. B. 49 199–208.

    MATH  MathSciNet  Google Scholar 

  • Reid, N. (1996). Likelihood and higher-order approximations to tail areas: A review and annotated bibliography. Canad. J. Statist. 24 141–166.

    Article  MATH  MathSciNet  Google Scholar 

  • Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York.

    MATH  Google Scholar 

  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist 6 461–464.

    Article  MATH  MathSciNet  Google Scholar 

  • Shun, Z. and McCullagh, P. (1995). Laplace approximation of high-dimensional integrals. J. Roy. Statist. Soc. Ser. B 57 749–760.

    MATH  MathSciNet  Google Scholar 

  • Skinner, L. A. (1980). Note on the asymptotic behavior of multidimensional Laplace integrals. SIAM J. Math. Anal. 11 911–917.

    Article  MATH  MathSciNet  Google Scholar 

  • Talagrand, M. (1987). The Glivenko–Cantelli problem. Ann. Probab. 15 837–870.

    Article  MATH  MathSciNet  Google Scholar 

  • Temme, N. M. (1982). The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions. SIAM J. Math. Anal. 13 239–253.

    Article  MATH  MathSciNet  Google Scholar 

  • Temme, N. M. (1987). Incomplete Laplace integrals: Uniform asymptotic expansion with application to the incomplete beta function. SIAM J. Math. Anal. 18 1638–1663.

    Article  MATH  MathSciNet  Google Scholar 

  • Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81 82–86.

    Article  MATH  MathSciNet  Google Scholar 

  • Tierney, L., Kass, R. E. and Kadane, J. B. (1989). Approximate marginal densities of nonlinear functions. Biometrika 76 425–433. [Correction (1991) 78 233–234.]

    Article  MATH  MathSciNet  Google Scholar 

  • van der Vaart, A. W. and Wellner, J. A. (2000). Preservation theorems for Glivenko–Cantelli and uniform Glivenko–Cantelli classes. In High Dimensional Probability II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 115–133. Birkhäuser, Boston.

    Google Scholar 

  • Walker, A. M. (1969). On the asymptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80–88.

    MATH  MathSciNet  Google Scholar 

  • Wall, H. S. (1948). Analytic Theory of Continued Fractions. Van Nostrand, New York.

    MATH  Google Scholar 

  • Wong, R. (1973). On uniform asymptotic expansion of definite integrals. J. Approx. Theory 7 76–86.

    Article  MATH  Google Scholar 

  • Wong, R. (1989). Asymptotic Approximations of Integrals. Academic Press, New York.

    MATH  Google Scholar 

  • Woodroofe, M. (1992). Integrable expansions for posterior distributions for one-parameter exponential families. Statist. Sinica 2 91–111.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Room 2-245, Cambridge, Massachusetts, 02139, USA

    R. M. Dudley

  2. Department of Mathematical Sciences, Bentley College, 175 Forest ST., Waltham, Massachusetts, 02154, USA

    D. Haughton

Authors
  1. R. M. Dudley
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  2. D. Haughton
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Correspondence to R. M. Dudley .

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Editors and Affiliations

  1. , Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, USA

    Evarist Giné

  2. , Mathematics, Georgia Institute of Technology, Atlanta, 30332-0160, USA

    Vladimir Koltchinskii

  3. Inst. Mathematics & Informatics, Akademijos 4, Vilnius, LT-08663, Lithuania

    Rimas Norvaisa

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Dudley, R.M., Haughton, D. (2010). Asymptotic Normality with Small Relative Errors of Posterior Probabilities of Half-Spaces. In: Giné, E., Koltchinskii, V., Norvaisa, R. (eds) Selected Works of R.M. Dudley. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5821-1_29

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