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On inadmissibility of some unbiased estimates of loss. (English) Zbl 0647.62018

Statistical decision theory and related topics IV, Pap. 4th Purdue Symp., West Lafayette/Indiana 1986, Vol. 1, 361-379 (1988).
[For the entire collection see Zbl 0638.00031.]
One is often interested in the discrepancy between an unknown (vector) parameter and its point estimate. This discrepancy (or “loss”) is itself unknown, being a function of both parameter and data. When the data is an observation on a p-variate normal distribution with identity covariance matrix, Stein’s unbiased estimate of risk is a natural unbiased estimator of this loss. How good is this unbiased loss estimator, when judged, say, by squared error loss?
In summary, this paper shows that if the point estimate is the maximum likelihood estimate, the unbiased loss estimator is admissible for \(p\leq 4(!)\), and inadmissible for \(p\geq 5\). An “improved” loss estimator is derived, which can achieve gains on the order of 20 %. It is given an empirical Bayes interpretation, and shown to be empirical Bayes minimax. When the point estimate is the James-Stein estimator, the unbiased loss estimator is again inadmissible for \(p\geq 5\), and larger gains of the order of 50-60 % are possible.

MSC:

62C15 Admissibility in statistical decision theory
62C12 Empirical decision procedures; empirical Bayes procedures
62C10 Bayesian problems; characterization of Bayes procedures
62H12 Estimation in multivariate analysis

Citations:

Zbl 0638.00031