Variance estimation with diffuse prior information

This paper deals with the problem of estimating the variance based on a random sample from a normal distribution with mean [mu] and variance [sigma] = 1/[tau]. In approaching the problem from a Bayesian point of view, a natural conjugate family of priors for ([mu], [tau]) consists of [tau] having a gamma distribution and the conditional distribution of [mu] given [tau] being normal with precision [delta][tau] for [delta] > 0 and known. This problem is discussed in the situation where we have no, or choose not to elicit any, prior information about the parameters [mu] and [tau]. It is shown that under various schemes of diffuse prior information of [tau], the obtained Bayes estimates for [sigma]2 turn out to be inferior in a mean squared error sense to the known classical estimates. In particular, a uniform prior on [sigma]2 = 1/[tau] yields the worst estimate among the ones under consideration.