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September, 1955 Ordered Families of Distributions
E. L. Lehmann
Ann. Math. Statist. 26(3): 399-419 (September, 1955). DOI: 10.1214/aoms/1177728487


A comparison is made of several definitions of ordered sets of distributions, some of which were introduced earlier by the author [7], [8] and by Rubin [10]. These definitions attempt to make precise the intuitive notion that large values of the parameter which labels the distributions go together with large values of the random variables themselves. Of the various definitions discussed the combination of two, (B) and (C) of Section 2, appears to be statistically most meaningful. In Section 3 it is shown that this ordering implies monotonicity for the power function of sequential probability ratio tests. In Section 4 the results are applied to obtaining tests that give a certain guaranteed power with a minimum number of observations. Finally, in Section 5, certain consequences are derived regarding the comparability of experiments in the sense of Blackwell [1].


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E. L. Lehmann. "Ordered Families of Distributions." Ann. Math. Statist. 26 (3) 399 - 419, September, 1955.


Published: September, 1955
First available in Project Euclid: 28 April 2007

zbMATH: 0065.11906
MathSciNet: MR71684
Digital Object Identifier: 10.1214/aoms/1177728487

Rights: Copyright © 1955 Institute of Mathematical Statistics

Vol.26 • No. 3 • September, 1955
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