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].
Received September 14, 1954
This paper was prepared with the partial support of the Office of Naval Research.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
D. Blackwell, “Comparison of experiments,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. California Press, Berkeley, 1951, pp. 93-102.
D. Blackwell, “Equivalent comparisons of experiments,” Ann. Math. Stat., Vol. 24 (1953), pp. 265-272.
D. R. Cox, “Sequential tests for composite hypotheses,” Proc. Cambridge Philos. Soc, Vol. 48 (1952), pp. 290-299.
N. L. Johnson, “Some notes on the application of sequential methods in the analysis of variance,” Ann. Math. Stat., Vol. 24 (1953), pp. 614-623.
J. L. Hodges, Jr. and E. L. Lehmann, “Testing the approximate validity of statistical hypotheses,” J. Roy. Stat. Soc., to be published.
W. Kruskal, “The monotonicity of the ratio of two noncentral t density functions,” Ann. Math. Stat., Vol. 25 (1954), pp. 162-165.
E. L. Lehmann, “Consistency and unbiasedness of certain nonparametric tests,” Ann. Math. Stat., Vol. 22 (1951), pp. 165-179.
E. L. Lehmann, “Testing multiparameter hypotheses,” Ann. Math. Stat., Vol. 23 (1952), pp. 541-552.
E. L. Lehmann, “Some principles of the theory of testing hypotheses,” Ann. Math. Stat., Vol. 20 (1950), pp. 1-26.
H. Rubin, “A complete class of decision procedures for distributions with monotone likelihood ratio,” Ann. Math. Stat., Vol. 22 (1951), p. 608 (Abstract).
I. J. Schoenberg, “On Polya frequency functions. I. The totally positive functions and their Laplace transforms,” J. d’Analyse Mathematique, Vol. 1 (1951), pp. 331-374.
A. Wald, Sequential Analysis, John Wiley and Sons, New York, 1947.
A. Wald, Statistical Decision Functions, John Wiley and Sons, New York, 1950.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
This chapter is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
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. https://doi.org/10.1007/978-1-4614-1412-4_60
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1412-4_60
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4614-1411-7
Online ISBN: 978-1-4614-1412-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)