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December 1995 Posterior convergence given the mean
B. Clarke, J. K. Ghosh
Ann. Statist. 23(6): 2116-2144 (December 1995). DOI: 10.1214/aos/1034713650


For various applications one wants to know the asymptotic behavior of $w(\theta | \overline{X})$, the posterior density of a parameter $\theta$ given the mean $\overline{X}$ of the data rather than the full data set. Here we show that $w(\theta | \overline{X})$ is asymptotically normal in an $L^1$ sense, and we identify the mean of the limiting normal and its asymptotic variance. The main results are first proved assuming that $X_1, \dots, X_n, \dots$ are independent and identical; suitable modifications to obtain results for the nonidentical case are given separately. Our results may be used to construct approximate HPD (highest posterior density) sets for the parameter which is of use in the statistical theory of standardized educational tests. They may also be used to show the covariance between two test items conditioned on the mean is asymptotically nonpositive. This has implications for constructing tests of item independence.


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B. Clarke. J. K. Ghosh. "Posterior convergence given the mean." Ann. Statist. 23 (6) 2116 - 2144, December 1995.


Published: December 1995
First available in Project Euclid: 15 October 2002

zbMATH: 0854.62023
MathSciNet: MR1389868
Digital Object Identifier: 10.1214/aos/1034713650

Primary: 62F15
Secondary: 62E20

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 6 • December 1995
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