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Asymptotic Expansions of Posterior Distributions in Nonregular Cases

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Abstract

We study the asymptotic behaviour of the posterior distributions for a one-parameter family of discontinuous densities. It is shown that a suitably centered and normalized posterior converges almost surely to an exponential limit in the total variation norm. Further, asymptotic expansions for the density, distribution function, moments and quantiles of the posterior are also obtained. It is to be noted that, in view of the results of Ghosh et al. (1994, Statistical Decision Theory and Related Topics V, 183-199, Springer, New York) and Ghosal et al. (1995, Ann. Statist., 23, 2145-2152), the nonregular cases considered here are essentially the only ones for which the posterior distributions converge. The results obtained here are also supported by a simulation experiment.

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Authors and Affiliations

  1. Division of Theoretical Statistics and Mathematics, Indian Statistical Institute, 203 B.T. Road, Calcutta-, 700035, India

    Subhashis Ghosal

  2. Computer Science Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta-, 700035, India

    Tapas Samanta

Authors
  1. Subhashis Ghosal
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  2. Tapas Samanta
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Ghosal, S., Samanta, T. Asymptotic Expansions of Posterior Distributions in Nonregular Cases. Annals of the Institute of Statistical Mathematics 49, 181–197 (1997). https://doi.org/10.1023/A:1003127108965

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  • DOI: https://doi.org/10.1023/A:1003127108965

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