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Introduction to Kolmogorov (1933) On the Empirical Determination of a Distribution

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Breakthroughs in Statistics

Part of the book series: Springer Series in Statistics ((PSS))

Abstract

In 1933, A.N. Kolmogorov (1933a) published a short but landmark paper in the Italian Giornale dell’Istituto Italiano degli Attuari. He formally defined the empirical distribution function (EDF) and then enquired how close this would be to the true distribution F(x) when this is continuous. This leads naturally to the definition of what has come to be known as the Kolmogorov statistic (or sometimes the Kolmogorov- Smirnov statistic) D, and Kolmogorov not only then demonstrates that the difference between the EDF and F(x) can be made as small as we please as the sample size n becomes larger, but also gives a method for calculating the distribution of D at specified points, for finite n, and uses this to give the asymptotic distribution of D. The ideas in this paper have formed a platform for a vast literature, both of interesting and important probability problems, and also concerning methods of using the Kolmogorov statistic (and other statistics) for testing fit to a distribution. This literature continues with great strength today, after over 50 years, showing no signs of diminishing. It is evident that the ideas set in motion by Kolmogorov are of paramount importance in statistical analysis, and variations on the probabilistic problems, including modern methods of treating them, continue to hold attention.

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Authors and Affiliations

  1. Simon Fraser University, Canada

    M. A. Stephens

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  1. M. A. Stephens
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Editors and Affiliations

  1. College of Business and Management, University of Maryland at College Park, 20742, College Park, MD, USA

    Samuel Kotz

  2. Department of Statistics Phillips Hall, The University of North Carolina at Chapel Hill, 27599, Chapel Hill, NC, USA

    Norman L. Johnson

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© 1992 Springer-Verlag New York, Inc.

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Stephens, M.A. (1992). Introduction to Kolmogorov (1933) On the Empirical Determination of a Distribution. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4380-9_9

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  • DOI: https://doi.org/10.1007/978-1-4612-4380-9_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94039-7

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