Statistics – Methodology
Scientific paper
2012-03-26
Statistical Science 2012, Vol. 27, No. 1, 115-134
Statistics
Methodology
Published in at http://dx.doi.org/10.1214/11-STS363 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M
Scientific paper
10.1214/11-STS363
This review traces the evolution of theory that started when Charles Stein in 1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197--206, Univ. California Press] showed that using each separate sample mean from $k\ge3$ Normal populations to estimate its own population mean $\mu_i$ can be improved upon uniformly for every possible $\mu=(\mu_1,...,\mu_k)'$. The dominating estimators, referred to here as being "Model-I minimax," can be found by shrinking the sample means toward any constant vector. Admissible minimax shrinkage estimators were derived by Stein and others as posterior means based on a random effects model, "Model-II" here, wherein the $\mu_i$ values have their own distributions. Section 2 centers on Figure 2, which organizes a wide class of priors on the unknown Level-II hyperparameters that have been proved to yield admissible Model-I minimax shrinkage estimators in the "equal variance case." Putting a flat prior on the Level-II variance is unique in this class for its scale-invariance and for its conjugacy, and it induces Stein's harmonic prior (SHP) on $\mu_i$.
Lysy Martin
Morris Carl N.
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