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Ordered Families of Distributions

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Part of the Selected Works in Probability and Statistics book series (SWPS)

Summary and introduction

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].


  • Power Function
  • Exponential Family
  • Joint Density
  • Minimum Power
  • Intuitive Notion

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Received September 14, 1954

This paper was prepared with the partial support of the Office of Naval Research.


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Lehmann, E.L. (2012). Ordered Families of Distributions. In: Rojo, J. (eds) Selected Works of E. L. Lehmann. Selected Works in Probability and Statistics. Springer, Boston, MA.

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