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Statistical Tests: Verifying Hypotheses

  • Simon ŠircaEmail author
Chapter
Part of the Graduate Texts in Physics book series (GTP)

Abstract

Statistical tests based on hypotheses are used to statistically verify or disprove, at a certain level of significance, models of populations and their probability distributions. The null and alternative hypothesis are the corner-stones of each such verification, and go hand-in-hand with the possibility of inference errors; these are defined first, followed by the exposition of standard parametric tests for normally distributed variables (tests of mean, variance, comparison of means, variances). Pearson’s \(\chi ^2\)-test is introduced as a means to ascertain the quality of regression (goodness of fit) in the case of binned data. The Kolmogorov–Smirnov test with which binned data can be compared to a continuous distribution function or two binned data sets can be compared to each other, is discussed as a distribution-free alternative.

References

  1. 1.
    W.J. Conover, Practical Nonparametric Statistics, 3rd edn. (Wiley, New York, 1999)Google Scholar
  2. 2.
    F. James, Statistical Methods in Experimental Physics, 2nd edn. (World Scientific, Singapore, 2006)zbMATHGoogle Scholar
  3. 3.
    A. Kolmogorov, Sulla determinazione empirica di una legge di distribuzione, Giornalo dell’Istituto Italiano degli Attuari 4, 461 (1933). Translated in A.N. Shiryayev (Ed.), Selected works of A.N. Kolmogorov, Vol. II (Springer Science+Business Media, Dordrecht, 1992) p. 139Google Scholar
  4. 4.
    N. Smirnov, Sur les écarts de la courbe de distribution empirique. Rec. Math. 6, 3 (1939)zbMATHGoogle Scholar
  5. 5.
    S. Facchinetti, A procedure to find exact critical values of Kolmogorov–Smirnov test, Statistica Applicata—Ital. J. Appl. Stat. 21, 337 (2009)Google Scholar
  6. 6.
    J.W. Pratt, J.D. Gibbons, Concepts of Nonparametric Theory (Springer, New York, 1981)CrossRefzbMATHGoogle Scholar
  7. 7.
    M.A. Stephens, Tests based on EDF statistics, in Goodness of Fit Techniques, ed. by R.B. D’Agostino, M.A. Stephens (Marcel Dekker, New York, 1986), pp. 97–194Google Scholar
  8. 8.
    T.W. Anderson, D.A. Darling, Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann. Math. Statist. 23, 193 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    T.W. Anderson, D.A. Darling, A test of goodness of fit. J. Am. Stat. Assoc. 49, 765 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M.A. Stephens, EDF statistics for goodness of fit and some comparisons. J. Am. Stat. Assoc. 69, 730 (1974)CrossRefGoogle Scholar
  11. 11.
    M.A. Stephens, Asymptotic results for goodness-of-fit statistics with unknown parameters. Annals Stat. 4, 357 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M.A. Stephens, Goodness of fit for the extreme value distribution. Biometrika 64, 583 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M.A. Stephens, Goodness of fit with special reference to tests for exponentiality, Technical Report No. 262, Department of Statistics, Stanford University, Stanford, 1977Google Scholar
  14. 14.
    M.A. Stephens, Tests of fit for the logistic distribution based on the empirical distribution function. Biometrika 66, 591 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A.C. Davison, D.V. Hinkley, Bootstrap Methods and their Application (Cambridge University Press, Cambridge, 1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    A.G. Frodesen, O. Skjeggestad, H. Tøfte, Probability and statistics in particle physics (Universitetsforlaget, Bergen, 1979)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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