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The split-up algorithm: a fast symbolic method for computing p-values of distribution-free statistics

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Summary

Many distribution-free statistics have the drawback that computing exact p-values under the null hypothesis is an intensive task. When the sample sizes are small or the number of ties is large, approximations are often unsatisfactory. Moreover, tables of exact critical values are not available for conditional rank statistics (ties, censoring), for rank statistics with arbitrary regression constants, or for permutation test statistics. In those cases, it is important to have a fast algorithm for computing exact p-values. We present a new algorithm and apply it to a large class of distribution-free one-sample, two-sample and serial statistics. The algorithm is based on splitting the probability generating function of the test statistic into two parts. We compare the speed of this “split-up algorithm” to that of existing procedures and we conclude that our new algorithm is faster in many cases.

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References

  • Gibbons, J.D., and J.W. Pratt (1975). P-values: interpretation and methodology. American Statistician, 29, 20–25.

    MATH  Google Scholar 

  • Good, P.I. (1994). Permutation Tests, a Practical Guide to Resampling Methods for Testing Hypotheses. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Hallin, M., and G. Mélard (1988). Rank-based tests for randomness against first-order serial dependence. Journal of the American Statistical Association, 83, 1117–1128.

    Article  MathSciNet  Google Scholar 

  • Hallin, M., and M.L. Puri (1988). Optimal rank-based procedures for time series analysis: testing an ARMA model against other ARMA models. Annals of Statistics, 16, 402–432.

    Article  MathSciNet  Google Scholar 

  • Hallin, M., and M.L. Puri (1991). Time series analysis via rank order theory: signed-rank tests for ARMA models. Journal of Multivariate Analysis, 39, 1–29.

    Article  MathSciNet  Google Scholar 

  • Kendall, M.G., and A. Stuart (1977). The Advanced Theory of Statistics, Volume 2. Charles Griffin & Co., London.

    MATH  Google Scholar 

  • Mehta, C.R., N.R. Patel, and L.J. Wei (1988). Constructing exact significance tests with restricted randomization rules. Biometrika, 75, 295–302.

    Article  Google Scholar 

  • Pagano, M., and D. Tritchler (1983). On obtaining permutation distributions in polynomial time. Journal of the American Statistical Association, 78, 435–441.

    Article  MathSciNet  Google Scholar 

  • Prentice, R.L., and P. Marek (1979). A qualitative discrepancy between censored data rank tests. Biometrics, 35, 861–867.

    Article  Google Scholar 

  • Puri, M.L., and P.K. Sen (1985). Nonparametric Methods in General Linear Models. Wiley, New York.

    MATH  Google Scholar 

  • Streitberg, B., and J. Röhmel (1986). Exact distributions for permutation and rank tests: an introduction to some recently published algorithms. Statistical Software Newsletter, 12, 10–17.

    Google Scholar 

  • Streitberg, B., and J. Röhmel (1987). Exakte Verteilungen für Rang-und Randomisierungstests im allgemeinen c-Stichprobenproblem. EDV in Medizin und Biologie, 18, 12–19.

    Google Scholar 

  • Van de Wiel, M.A. (1996). Computing exact distributions of rank statistics with computer algebra. Master’s thesis, Eindhoven University of Technology, The Netherlands.

    Google Scholar 

  • Van de Wiel, M.A. (1998). Exact distributions of two-sample rank statistics and block rank statistics using computer algebra. Technical Report Memorandum COSOR 98-14, Eindhoven University of Technology, The Netherlands.

  • Van de Wiel, M.A., and A. Di Bucchianico (2001). Fast computation of the exact null distribution of Spearman’s rho and Page’s Lstatistic for samples with and without ties. Journal of Statistical Planning and Inference, 92, 133–145.

    Article  MathSciNet  Google Scholar 

  • Wald, A., and J. Wolfowitz (1943). An exact test for randomness in the nonparametric case based on serial correlation. Annals of Mathematical Statistics, 14, 378–388.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank Alessandro Di Bucchianico and the referees of this journal for their useful comments on earlier versions of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Mark van de Wiel

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  1. Mark van de Wiel
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Correspondence to Mark van de Wiel.

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van de Wiel, M. The split-up algorithm: a fast symbolic method for computing p-values of distribution-free statistics. Computational Statistics 16, 519–538 (2001). https://doi.org/10.1007/s180-001-8328-6

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