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

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Information Systems Architecture and Technology: Proceedings of 37th International Conference on Information Systems Architecture and Technology – ISAT 2016 – Part III

Part of the book series: Advances in Intelligent Systems and Computing ((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.

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References

  1. Bukietyńska A: Kombinatoryczny test inwersji, Metody Ilosciowe w ekonomii, WSB Poznan, pp. 152162, (2008)

    Google Scholar 

  2. David, F.N., Kendall, M.G., Barton, D.E.: Symmetric Function and Allied Tables, p. 241. Cambridge (1966)

    Google Scholar 

  3. Feller, W.: An Introduction to Probability Theory and its Application. Wiley, New York, London (1961)

    Google Scholar 

  4. Ferguson, S., Genest, Ch., Hallin, M.: Kendall’s tau for autocorrelation. Can. J. Stat. 28, 587–604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ferguson, S., Genest, Ch., Hallin, M.: Kendall’s tau for autocorrelation. Dep. Stat. Pap., UCLA (2011)

    MATH  Google Scholar 

  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. Janjic, M.: A generating function for numbers of insets. J. Integer Seq. 17 #14.9.7 (2014)

    Google Scholar 

  8. Kendall, M.G., Buckland, W.R.: A Dictionary of Statistical Terms. OLIVER AND BOYD, Edinburgh, London (1960)

    MATH  Google Scholar 

  9. Netto E.: Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, p. 96. (1927)

    Google Scholar 

  10. The On-Line Encyclopedia of Integer Sequences, sequence A008302

    Google Scholar 

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Author information

Authors and Affiliations

  1. Wroclaw School of Banking Ul, Fabryczna 29-31, 53-609, Wroclaw, Poland

    Mariusz Czekala & Agnieszka Bukietyńska

Authors
  1. Mariusz Czekala
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  2. Agnieszka Bukietyńska
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Corresponding author

Correspondence to Mariusz Czekala .

Editor information

Editors and Affiliations

  1. Faculty of Computer Science and Management, Department of Computer Science, Wrocław University of Technology, Wrocław, Poland

    Jerzy Świątek

  2. Department of Management Systems,Faculty of Computer Science and Management, Wrocław University of Technology, Wrocła Department of Management Systems, Wrocław, Poland

    Zofia Wilimowska

  3. Faculty of Computer Science and Management,, Wrocław University of Technology Department of IT and Management, Wrocław, Poland

    Leszek Borzemski

  4. Faculty of Computer Science and Management,, Wrocław University of Technology Department of Computer Science, Wrocław, Poland

    Adam Grzech

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Czekala, M., Bukietyńska, A. (2017). Distribution of Inversions and the Power of the τ- Kendall’s Test. In: Świątek, J., Wilimowska, Z., Borzemski, L., Grzech, A. (eds) Information Systems Architecture and Technology: Proceedings of 37th International Conference on Information Systems Architecture and Technology – ISAT 2016 – Part III. Advances in Intelligent Systems and Computing, vol 523. Springer, Cham. https://doi.org/10.1007/978-3-319-46589-0_14

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  • DOI: https://doi.org/10.1007/978-3-319-46589-0_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-46588-3

  • Online ISBN: 978-3-319-46589-0

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