ASYMPTOTIC EXPANSIONS OF POSTERIOR DISTRIBUTIONS IN NONREGULAR CASES

SUBHASHIS GHOSAL^{1} AND TAPAS SAMANTA^{2}

1Division of Theoretical Statistics and Mathematics, Indian Statistical Institute,

203 B.T. Road, Calcutta-700035, India

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

(Received May 19, 1995; revised April 23, 1996)

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.

Key words and phrases: Asymptotic expansion, posterior distributions, nonregular cases.