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Bayes Set Estimators for the Mean of a Multivariate Normal Distribution and Rate of Convergence of Their Posterior Risk

Sudhakar Kunte and R. N. Rattihalli
Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)
Vol. 51, No. 1 (Feb., 1989), pp. 94-105
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25050726
Page Count: 12
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Bayes Set Estimators for the Mean of a Multivariate Normal Distribution and Rate of Convergence of Their Posterior Risk
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Abstract

In this paper we have obtained fixed sample size Bayes set estimators for the mean of a multivariate normal distribution (with precision matrix known) by using conjugate prior distribution of DeGroot (1970). The loss function is considered to be a linear combination of a measure of size of the set and the indicator of the non-coverage. Two measures of the size of a set considered are: (i) λ, the Lebesgue measure and (ii) r, the radius. We have also considered the restricted problems where the choice of the sets (action space) is restricted to (i) rectangles, (ii) spheres. We can also consider the Bayesian sequential version of these problems and obtain asymptotically optimal (A.O.) and asymptotically pointwise optimal (A.P.O.) stopping rules. For this purpose we can use the results of Gleser and Kunte (1976). If $\rho _{n}$ denotes the posterior risk of Bayes rule for sample size n, then we have to obtain a function f(n) such that $\rho _{n}$ is O(1/f(n)). We have obtained such functions f(n) in different cases and with these f(n) the results of Gleser and Kunte become directly applicable. The univariate analogue of the results obtained here are already available in Gleser and Kunte (1976).

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