How Much Better Are Better Estimators of a Normal Variance

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Abstract
This article considers the estimation problem of a normal variance σ2 on the basis of a random sample x 1, …,x n under quadratic loss when the mean ξ is unknown. The best equivariant estimator δ0 = Σ (xj – [xbar])2/(n + 1) is known to be inadmissible, but the extent to which this inadmissibility phenomenon is serious has not previously been considered. Herein, the mean squared error of the minimax and admissible estimator due to Brewster and Zidek (1974) is evaluted and an explicit formula for that estimator is given. It is shown that this risk function has maximum at ξ = 0, which is the mode of the corresponding generalized prior density. Locally optimal shrinkage scale-equivariant estimators are introduced and their risks are calculated for several values of the sample size n. It is observed that the Brewster–Zidek estimator has risk function close to that of locally optimal minimax shrinkage estimators, but that the latter cannot give more than 4% relative improvement on the traditional procedure in the sense that the ratio [(mean squared error of δ0) – (mean squared error of any shrinkage estimator)] ÷ (mean squared error of δ0) does not exceed .04 for all parametric values.

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