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A Class of Schur Procedures and Minimax Theory for Subset Selection

Jan F. Bjørnstad
The Annals of Statistics
Vol. 9, No. 4 (Jul., 1981), pp. 777-791
Stable URL: http://www.jstor.org/stable/2240846
Page Count: 15
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A Class of Schur Procedures and Minimax Theory for Subset Selection
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Abstract

The problem of selecting a random subset of good populations out of k populations is considered. The populations Π1, ⋯, Πk are characterized by the location parameters θ1, ⋯, θk and Πi is said to be a good population if $\theta_i > \max_{1 \leq j\leq k}\theta_j - \Delta$, and a bad population if θi ≤ max1 ≤ j ≤ k θj - Δ, where Δ is a specified positive constant. A theory for a special class of procedures, called Schur procedures, is developed, and applied to certain minimax problems. Subject to controlling the minimum expected number of good populations selected or the probability that the best population is in the selected subset, procedures are derived which minimize the expected number of bad populations selected or some similar criterion. For normal populations it is known that the classical "maximum-type" procedures has certain minimax properties. In this paper, two other procedures are shown to have several minimax properties. One is the "average-type" procedure. The other procedure has not previously been considered as a serious contender.

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