On Some Bayesian Solutions of the Neyman-Scott Problem

TLDR: In this article, a class of hierarchical Bayes (HB) estimators of σ 2 is introduced, and a subclass, each member of which dominates the best multiple of S (S being the error sum of squares) under the relative squared error loss L(a, σ2) = (a σ -2 -1)2.

Abstract: One of the two celebrated examples of Neyman and Scott (1948) is that in a fixed effects one-way analysis of variance model with normal homoscedastic errors, the maximum likelihood estimator of the error variance, say σ 2 is inconsistent as the cell-size remains fixed, but the number of cells grows to infinity. The UMVUE, or the best multiple estimator of the error sum of squares does not suffer from this drawback. However, the best multiple estimator is inadmissible, as it is dominated by a Stein-type estimator. The Stein-estimator, on the other hand, being non-smooth, is itself inadmissible. The present paper introduces a class of hierarchical Bayes (HB) estimators of σ 2, and identifies a subclass, each member of which dominates the best multiple of S (S being the error sum of squares) under the relative squared error loss L(a, σ 2) = (a σ -2 -1)2. Included in our class is the Brewster-Zidek (1974) estimator of σ 2. Also, we have provided analytic expressions for the risk improvement of the HB estimators over the best multiple estimator. Numerical values of the percentage risk improvement are given in some special cases. These calculations indicate that the risk-improvement over the best multiple estimator can often be quite substantial.