Bayesian Estimation of the Normal Location Model: A Non-Standard Approach

We consider the estimation of the location parameter $\theta$ in the normal location model and study the sampling properties of shrinkage estimators derived from a non-standard Bayesian approach that places the prior on a scaled version of $\theta$, interpreted as the “population $t$-ratio.” We show that the finite-sample distribution of these estimators is not centred at $\theta$ and is generally non-normal. In the asymptotic theory, we prove uniform $\sqrt{n}$-consistency of our estimators and obtain their asymptotic distribution under a general moving-parameter setup that includes both the fixed-parameter and the local-parameter settings as special cases.