Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T11:52:45.176Z Has data issue: false hasContentIssue false
  • English
  • Français

The Estimation of the Parameters of a Negative Binomial Distribution with Special Reference to Psychological Data

Published online by Cambridge University Press:  01 January 2025

Herbert S. Sichel
Herbert S. Sichel*
Affiliation:
National Institute for Personnel Research, South African Council for Scientific and Industrial Research
Get access Rights & Permissions [Opens in a new window]

Abstract

As an analytical tool the negative binomial distribution may have wide applications in the psychological field. The estimation of its parameters is demonstrated to be often inefficient when fitting by the method of moments. This causes possibly true hypotheses to be rejected. Formulas for the efficiency of the moment method and solution of the likelihood equations are derived. Efficiency graphs and detailed tables for the \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda (r,\hat p)$$\end{document} function reduce the maximum-likelihood method to a minimum of computational labour. Practical applications of the ease and power of the M.L. procedure are given.

Type
Original Paper
Information
Psychometrika , Volume 16 , Issue 1 , March 1951 , pp. 107 - 127
Copyright
Copyright © 1951 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

The author wishes to express his gratitude to the South African Council for Scientific and Industrial Research for permission to publish this paper; to Mr. A. G. Arbous for supplying the data on absenteeism; to Mr. R. V. Sutton for the data on Two-hand Co-ordination errors; and last, but not least, to the staff of the Statistical Section who computed various tables under the expert guidance of Mr. J. S. Maritz.

References

Fisher, R. A. On the mathematical foundations of theoretical statistics. Philos. Trans., 1921, A222, 309–309.Google Scholar
Greenwood, M., and Yule, G. U. An inquiry into the nature of frequency-distributions of multiple happenings, etc.. J. roy. stat. Soc., 1920, 83, 255–255.CrossRefGoogle Scholar
Kendall, M. G. The advanced theory of statistics, London: Charles Griffin and Co., Ltd., 1945.Google Scholar
Fisher, R. A. The negative binomial distribution. Ann. Eug., Lond., 1941, 11, 182–182.CrossRefGoogle Scholar
Haldane, J. B. S. The fitting of binomial distributions. Ann. Eug., Lond., 1941, 11, 179–179.CrossRefGoogle Scholar
Maritz, J. S. On the validity of inferences drawn from the fitting of Poisson and negative binomial distributions to observed accident data. Psychol. Bull., 1950, 47, 434–434.CrossRefGoogle ScholarPubMed
Fisher, R. A. Statistical methods for research workers, Edinburgh: Oliver and Boyd, 1948.Google Scholar
Chambers, E. G., and Yule, G. U. Theory and observations in the investigation of accident causation. Supp. J. roy. stat. Soc., 1941, 7(2), 89–89.CrossRefGoogle Scholar