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Distribution of Inversions and the Power of the τ- Kendall’s Test

  • Mariusz Czekala
  • Agnieszka Bukietyńska
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 523)

Abstract

In this paper we firstly present the so far known result of the distribution of the number of inversions in the sequence of random variables. We say for the sequence \( (X_{1} , X_{2} , \ldots , X_{n} ) \) the inversion is given for i, j and \( X_{i} \), \( X_{j} \) when \( i < j \) and \( X_{i} > X_{j} \). Under independence we show the exact distribution of the number of inversions in the permutation (equivalent to τ- Kendall distribution). The difference is in normalizing constants. Considering inversions is more convenient. The aim of this paper is to provide the exact distribution respectively for the dependence case.

Keywords

Τ- kendall Dependence Permutation Inversion Power of the test 

References

  1. 1.
    Bukietyńska A: Kombinatoryczny test inwersji, Metody Ilosciowe w ekonomii, WSB Poznan, pp. 152162, (2008)Google Scholar
  2. 2.
    David, F.N., Kendall, M.G., Barton, D.E.: Symmetric Function and Allied Tables, p. 241. Cambridge (1966)Google Scholar
  3. 3.
    Feller, W.: An Introduction to Probability Theory and its Application. Wiley, New York, London (1961)Google Scholar
  4. 4.
    Ferguson, S., Genest, Ch., Hallin, M.: Kendall’s tau for autocorrelation. Can. J. Stat. 28, 587–604 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ferguson, S., Genest, Ch., Hallin, M.: Kendall’s tau for autocorrelation. Dep. Stat. Pap., UCLA (2011)zbMATHGoogle Scholar
  6. 6.
    Hallin, M., Metard, G.: Rank–based test for randomness against first order dependence. J. Am. Stat. Assoc. 83, 1117–1128 (1988)Google Scholar
  7. 7.
    Janjic, M.: A generating function for numbers of insets. J. Integer Seq. 17 #14.9.7 (2014)Google Scholar
  8. 8.
    Kendall, M.G., Buckland, W.R.: A Dictionary of Statistical Terms. OLIVER AND BOYD, Edinburgh, London (1960)zbMATHGoogle Scholar
  9. 9.
    Netto E.: Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, p. 96. (1927)Google Scholar
  10. 10.
    The On-Line Encyclopedia of Integer Sequences, sequence A008302Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Wroclaw School of Banking UlWroclawPoland

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