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A New Exact Runs Test for Randomness

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Computing Science and Statistics

Abstract

Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple form; we discuss the programming of these algorithms for values of n that are large, but not so large that asymptotic results can be invoked. Using the conditional distribution of Nn,k we derive a test for randomness and compare it with standard procedures based on runs, ranks, and variances. The simulation results showed that the new test is significantly more powerful in detecting certain types of clustering. Applications in neurology and reliability are provided.

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Bibliography

  • Agin, Marilyn A. A New Exact Test for Randomness based on the Binomial Distribution of Order k. M.S. thesis, Michigan Technological University, 1990.

    Google Scholar 

  • Barton, D. E. and F. N. David (1958a). “Nonrandomness in a Sequence of Two Alternatives (II. Runs Test),” Biometrika. 45, 253–256.

    MATH  Google Scholar 

  • (1958b). “A Test for Birth Order Effect,” Annals of Human Genetics, 22, (Vol. 1, No. 3), 250–257.

    Article  MATH  Google Scholar 

  • Burr, E.J. and Gwenda Cane (1961). “Longest run of consecutive observations having a specified attribute,” Biometrika. 48, 461–465.

    MathSciNet  MATH  Google Scholar 

  • Chiang, D. T. and S. C. Niu (1981). “Reliability of Consecutive-k-out-of-n:F System,” IEEE Transactions on Reliability. R-30 (No. 1, April), 87–89.

    Article  Google Scholar 

  • Chiang, D. T. and Ru-fang Chiang (1986). “Relayed Communication via consecutive k-out-of-n:F system,” IEEE Transactions on Reliability, R-35 (April), 65–67.

    Article  Google Scholar 

  • Conover, W. J., Thomas R. Bernent and Ronald L. Iman (1979). “On a Method for Detecting Clusters of Possible Uranium Deposits,” Technometrics. 21, (No. 3, August), 277–282.

    Article  Google Scholar 

  • Derman, Cyrus, Gerald J. Lieberman and Sheldon M. Ross (1982). “On the Consecutive k-out-of-n:F System,” IEEE Transactions on Reliability. R-31, 57–62.

    Article  Google Scholar 

  • Dixon, W. J. (1940). “A Criterion for Testing the Hypothesis that Two Samples are from the Same Population,” Annals of Mathematical Statistics, 11, 199–204.

    Article  MathSciNet  MATH  Google Scholar 

  • Dyck, Peter James, M.D., P.K. Thomas, M.D., Edward H. Lambert, M.D., and Richard Bunge, M.D. Peripheral Neuropathy. Second Edition. Volumes I and II. Philadelphia, 1984.

    Google Scholar 

  • Feller, William. An Introduction to Probability Theory and its Applications. Volume I. John Wiley and Sons, Inc., New York, 1968.

    MATH  Google Scholar 

  • Gibbons, Jean Dickinson. Nonparametric Statistical Inference. McGraw Hill, New York, 1971.

    MATH  Google Scholar 

  • Nonparametric Methods for Quantitative Analysis. Second Edition. Columbus, 1985.

    Google Scholar 

  • Godbole, Anant P. (1990a). “Specific Formulae for some success run distributions,” Statistics and Probability Letters. 10, 119–124.

    Article  MathSciNet  MATH  Google Scholar 

  • (1990b).“On hypergeometric and related distributions of order k,” Communications in Statistics. Theory, and Methods, A-19, 1291–1301.

    Google Scholar 

  • (1990c). “Degenerate and Poisson convergence criteria for success runs,” Statistics and Probability Letters. 10, 247–255.

    Google Scholar 

  • Griffith, W.S. (1986). “On Reliability of Consecutive-k-out-of-n Failure Systems and their Generalizations,” in Reliability and Quality Control. A.P. Basu, Ed., North Holland, Amsterdam, 157–165.

    Google Scholar 

  • Haldane, J.B.S., F.R.S., and Cedric A. B. Smith (1948). “A Simple Exact Test for Birth-Order Effect,” Annals of Eugenics. 14,(2), 117–124.

    Google Scholar 

  • Hirano, Katuomi (1986). “Some Properties of the Distributions of Order k” in A. N. Philippou et al. (eds.) Fibonacci Numbers and Their Applications. Reidel, Dordrecht.

    Google Scholar 

  • Larsen, Richard J., Charles L. Holmes, and Clark W. Heath, Jr. (1973). “A Statistical Test for Measuring Unimodal Clustering: A Description of the Test and of its Application to Cases of Acute Leukemia in Metropolitan Atlanta, Georgia,” Biometrics, 29, 301–309.

    Article  MathSciNet  Google Scholar 

  • Mood, A. M. (1940). “The Distribution Theory of Runs,” Annals of Mathematical Statistics. 11, 367–392.

    Article  MathSciNet  MATH  Google Scholar 

  • Mosteller, F. (1941). “Note on an Application of Runs to Quality Control Charts,” Annals of Mathematical Statistics, 12, 228–232.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Brien, Peter C. (1976). “A Test for Randomness,” Biometrics. 32, 391–401.

    Article  MATH  Google Scholar 

  • O’Brien, Peter C. (1985). “A Runs Test Based on Runs Lengths,” Biometrics. 41, 237–244.

    Article  MathSciNet  MATH  Google Scholar 

  • Papastavridis, Stavros (1989). “The Number of Failed Components in a Consecutive-k-out-of-n:F System,” IEEE Transactions on Reliability. 38, 338–340.

    Article  MATH  Google Scholar 

  • (1990). “m-Consecutive-k-out-of-n:F Systems,” IEEE Transactions on Reliability. R39, 386–388.

    Google Scholar 

  • Philippou, Andreas N. and Frosso S. Makri (1986). “Successes, Runs and Longest Runs,” Statistics and Probability Letters. 4, 211–215.

    Article  MathSciNet  MATH  Google Scholar 

  • Swed, Frieda S. and C. Eisenhart (1943). “Tables for Testing Randomness of Grouping in a Sequence of Alternatives,” Annals of Mathematical Statistics. 14, 66–87.

    Article  MathSciNet  MATH  Google Scholar 

  • Wald, A. and J. Wolfowitz (1940). “On a Test Whether Two Samples are from the Same Population.” Annals of Mathematical Statistics. 11, 147–162.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Michigan Technological University, Houghton, Michigan, 49913, USA

    Marilyn A. Agin & Anant P. Godbole

Authors
  1. Marilyn A. Agin
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  2. Anant P. Godbole
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Editor information

Editors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Connie Page  & Raoul LePage  & 

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© 1992 Springer-Verlag New York, Inc.

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Agin, M.A., Godbole, A.P. (1992). A New Exact Runs Test for Randomness. In: Page, C., LePage, R. (eds) Computing Science and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2856-1_36

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  • DOI: https://doi.org/10.1007/978-1-4612-2856-1_36

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97719-5

  • Online ISBN: 978-1-4612-2856-1

  • eBook Packages: Springer Book Archive

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