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On the Translation Parameter Problem for Discrete Variables
The Annals of Mathematical Statistics
Vol. 22, No. 3 (Sep., 1951), pp. 393-399
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2236625
Page Count: 7
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For any chance variable x = (x1,⋯,xN) having known distribution, the translation parameter estimation problem is to estimate an unknown constant h, having observed y = (x1 + h,⋯,xN + h). Extending the work of Pitman , Girshick and Savage  have, for any loss function depending only on the error of estimate, described an estimate whose risk is a constant R independent of h, and have shown that under certain hypotheses their estimate is minimax. We investigate whether the Girshick-Savage estimate is admissible, i.e., whether it is impossible to find an estimate with risk R(h) ≤ R for all h and actual inequality for some h. We consider only bounded discrete variables x, and show that, if all values of x have all integer coordinates and if the loss f(d) from an error d is, for instance, strictly convex and assumes its minimum value, the Girshick-Savage estimate is admissible. Two examples in which the Girshick-Savage estimate is not admissible are given.
The Annals of Mathematical Statistics © 1951 Institute of Mathematical Statistics