Goodness-of-Fit Tests

  • Thorsten Dickhaus


Based on the substitution principle, we derive one-sample goodness-of-fit tests of Kolmogorov-Smirnov and Cramér-von Mises type, respectively. In the case of a completely specified null hypothesis, these tests are distribution-free, if the cumulative distribution function under the null is a continuous function. In the case of composite null hypotheses, we consider location-scale families, along with the maximum likelihood estimators of their parameters. In such cases, tests of Kolmogorov-Smirnov and Cramér-von Mises type are parameter-free and can be calibrated by means of computer simulations under an arbitrary distribution belonging to the null hypothesis.


  1. Anderson T, Darling D (1952) Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann Math Stat 23:193–212.
  2. Barndorff-Nielsen O, Cox D (1994) Inference and asymptotics. Chapman and Hall, LondonGoogle Scholar
  3. Deza MM, Deza E (2016) Encyclopedia of distances, 4th edn. Springer, Berlin.
  4. Kolmogorov A (1933) Sulla determinazione empirica di una legge di distribuzione. G Ist Ital Attuari 4:83–91zbMATHGoogle Scholar
  5. Shorack GR, Wellner JA (1986) Empirical processes with applications to statistics. Wiley series in probability and mathematical statistics. Wiley, New York, NYzbMATHGoogle Scholar
  6. Zehna P (1966) Invariance of maximum likelihood estimators. Ann Math Stat 37:744. MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Thorsten Dickhaus
    • 1
  1. 1.Institute for StatisticsUniversity of BremenBremenGermany

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