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Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity. (English) Zbl 1118.62309
Summary: We study consistency and asymptotic normality of posterior distributions of the natural parameter for an exponential family when the dimension of the parameter grows with the sample size. Under certain growth restrictions on the dimension, we show that the posterior distributions concentrate in neighborhoods of the true parameter and can be approximated by an appropriate normal distribution.

62E20 Asymptotic distribution theory in statistics
62F15 Bayesian inference
Full Text: DOI
[1] Bahadur, R.R, Some limit theorems in statistics, (1971), SIAM Pennsylvania · Zbl 0257.62015
[2] Berk, R.H, Consistency and asymptotic normality of MLE’s for exponential models, Ann. math. statist., 43, 193-204, (1972) · Zbl 0253.62005
[3] Bickel, P; Yahav, J, Some contributions to the asymptotic theory of Bayes solutions, Z. wahrsch. verw. gebiete, 11, 257-275, (1969) · Zbl 0167.17706
[4] Chritley, F; Marriott, P; Salmon, M, Preferred point geometry and statistical manifolds, Ann. statist., 21, 1197-1224, (1993) · Zbl 0798.62009
[5] Diaconis, P; Freedman, D, On the bernstein – von Mises theorem with infinite dimensional parameters, Technical report, (1997)
[6] M. Gasparini, Bayes Nonparametrics for biased sampling and density estimation, Unpublished Ph.D. thesis, University of Michigan, 1992.
[7] Ghosal, S, Normal approximation to the posterior distribution for generalized linear models with many covariates, Math. methods statist., 6, 332-348, (1997) · Zbl 0888.62071
[8] Ghosal, S, Asymptotic normality of posterior distributions in high-dimensional linear models, Bernoulli, 5, 315-331, (1999) · Zbl 0948.62007
[9] Ghosal, S; Ghosh, J.K; Ramamoorthi, R.V, Posterior consistency of Dirichlet mixtures in density estimation, Ann. statist., 27, 143-158, (1999) · Zbl 0932.62043
[10] Ghosal, S; Ghosh, J.K; Samanta, T, On convergence of posterior distributions, Ann. statist., 23, 2145-2152, (1995) · Zbl 0858.62024
[11] Ghosal, S; Ghosh, J.K; van der Vaart, A.W, Convergence rates of posterior distribution, Ann. statist., 28, (2000) · Zbl 1105.62315
[12] Grenander, U, Abstract inference, (1981), Wiley New York
[13] Haberman, S.J, Maximum likelihood estimates in exponential response models, Ann. statist., 5, 815-841, (1977) · Zbl 0368.62019
[14] Huber, P, Robust regression: asymptotics, conjectures, and Monte Carlo, Ann. statist., 1, 799-821, (1973) · Zbl 0289.62033
[15] Johnson, R.A, Asymptotic expansions associated with posterior distribution, Ann. math. statist., 42, 1241-1253, (1970)
[16] Le Cam, L, On some asymptotic properties of maximum likelihood estimates and related Bayes estimates, Univ. California publ. in stat., 1, 277-330, (1953)
[17] Pinsker, M.S, Optimal filtration of square integrable signals in Gaussian noise, Problems inform. transmission, 16, 120-133, (1980) · Zbl 0452.94003
[18] Portnoy, S, Asymptotic behavior of M-estimators of p regression parameters when p2/n is large. I: consistency, Ann. statist., 12, 1298-1309, (1984) · Zbl 0584.62050
[19] Portnoy, S, Asymptotic behavior of M-estimators of p regression parameters when p2/n is large. II: normal approximation, Ann. statist., 13, 1403-1417, (1985) · Zbl 0601.62026
[20] Portnoy, S, Asymptotic behavior of the empiric distribution of M-estimated residuals from a regression models with many parameters, Ann. statist., 14, 1152-1170, (1986) · Zbl 0612.62072
[21] Portnoy, S, Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity, Ann. statist., 16, 356-366, (1988) · Zbl 0637.62026
[22] Wong, W.H; Shen, X, Probability inequalities for likelihood ratios and convergence rates of sieve mles, Ann. statist., 23, 339-362, (1995) · Zbl 0829.62002
[23] Zhao, L.H, Bayesian aspects of some nonparametric problems, Technical report, (1998)
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