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A General Method of Obtaining Exact Sampling Probabilities of the Shannon-Wiener Measure of Information Ĥ

Published online by Cambridge University Press:  01 January 2025

James N. Cronholm
James N. Cronholm*
Affiliation:
US Army Medical Research Laboratory, Fort Knox, Kentucky
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Abstract

A two-variable generating function is described which yields the sampling probabilities of the Shannon-Wiener information measure. Expansion and collection of terms in like powers of the first variable imposes the restriction that the sum of the k category frequencies equal n; collection of terms in like powers of the second variable then produces terms whose coefficients are the required probabilities. The method may be used with either equal or unequal category probabilities for any finite n and k, and thus represents a general solution to the small sample problem. Tables of sampling probabilities are presented.

Type
Original Paper
Information
Psychometrika , Volume 28 , Issue 4 , December 1963 , pp. 405 - 413
Copyright
Copyright © 1963 The Psychometric Society

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Footnotes

*

The substance of this paper was reported at the meetings of the Southern Society for Philosophy and Psychology, Miami, Florida, April 12, 1963.

References

Miller, G. A. and Madow, W. G. On the maximum likelihood estimate of the Shannon-Wiener measure of information. Air Force Cambridge Research Center Technical Report 54–57, 1954.Google Scholar
Rogers, M. S. and Green, B. F. The moments of sample information when the alternatives are equally likely. In Quastler, H. (Eds.), Information theory in psychology. Glencoe, Ill.: The Free Press, 1955, 101–107.Google Scholar
Shannon, C. E. and Weaver, W. The mathematical theory of communication, Urbana: Univ. of Illinois Press, 1949.Google Scholar
Wiener, N. Cybernetics, New York: Wiley, 1948.Google ScholarPubMed