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Computational Statistics

, Volume 16, Issue 1, pp 97–107 | Cite as

Goodness-of-fit tests for the Cauchy distribution

  • Bora H. Onen
  • Dennis C. Dietz
  • Vincent C. Yen
  • Albert H. Moore
Article

Summary

This article presents several modified goodness-of-fit tests for the Cauchy distribution with unknown location and scale parameters. Monte Carlo studies are performed to calculate critical values for several tests based on the empirical distribution function. Power studies suggest that the modified Kuiper (V) test is the most powerful standard test against most alternate distributions over a full range of sample sizes. A reflection technique is also employed which yields substantial improvement in the power of this test against symmetric distributions.

Keywords

Cauchy distribution Goodness-of-fit Monte Carlo Kolmogorov-Smirnov test Kuiper test 

Notes

Acknowledgement

The authors are grateful to the anonymous reviewers, whose thoughtful suggestions have significantly improved the article.

References

  1. Andrews, D. F. (1972), Robust Estimates of Location, Princeton University Press, Princeton, NJ.Google Scholar
  2. Bai, Z. D. and Fu, J. C. (1987), “On the Maximum-Likelihood Estimator for the Location Parameter of a Cauchy Distribution,” The Canadian Journal of Statistics, 15, 137–146.MathSciNetCrossRefGoogle Scholar
  3. Barnett, V. D. (1966), “Evaluation of the Maximum Likelihood Estimator where the Likelihood Equation has Multiple Roots,” Biometrika, 53, 151–165.MathSciNetCrossRefGoogle Scholar
  4. Daniel, W. W. (1980), “Goodness-of-Fit: A Selected Bibliography for the Statistician and Researcher,” Public Administration Series: Bibliography, Vance Bibliographies.Google Scholar
  5. David, F. N. and Johnson, N. L. (1948), “The Probability Integral Transformation When Parameters Are Estimated From The Sample,” Biometrika, 35, 182–190.MathSciNetCrossRefGoogle Scholar
  6. Granger, C. W. J. and Orr, D. (1972), “Infinite Variance and Research Strategy in Time Series Analysis,” Journal of the American Statistical Association, 67, 275–285.zbMATHGoogle Scholar
  7. Haas, G., Bain, L., and Antle, C. (1970), “Inferences for the Cauchy Distribution Based on Maximum Likelihood Estimators,” Biometrika, 57: 403–408.zbMATHGoogle Scholar
  8. Harter, H. L. (1985), “Another Look at Plotting Positions,” Communications in Statistics, B14, 2, 317–343.CrossRefGoogle Scholar
  9. Higgins, J. J. and Tichenor, D. M. (1977), “Window Estimates of Location and Scale with Applications to the Cauchy Distribution,” Applied Mathematics and Computation, 3, 113–126.MathSciNetCrossRefGoogle Scholar
  10. Higgins, J. J. and Tichenor, D. M. (1978), “Efficiencies for Window Estimates of the Parameters of the Cauchy Distribution,” Applied Mathematics and Computation, 4, 157–165.CrossRefGoogle Scholar
  11. Johnson, N. L. and Kotz, S., and Balakrishnan, N. (1994), 2nd edn., Continuous Univariate Distributions-1, John Wiley and Sons, New York.zbMATHGoogle Scholar
  12. Lawless, J. F. (1982), Statistical Models and Methods for Lifetime Data Analysis, John Wiley and Sons, New York.zbMATHGoogle Scholar
  13. Meyer, S. L. (1989), Data Analysis for Scientists and Engineers, John Wiley and Sons, New York.Google Scholar
  14. Ocasio, F. (1985), A Modified Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises Test for the Cauchy Distribution with Unknown Location and Scale Parameters, MS thesis AFIT/ENC/GSO/85D-5, Air Force Institute of Technology.Google Scholar
  15. Shuster, E. F. (1973), “On the Goodness-of-Fit Problem for Continuous Symmetric Distributions,” Journal of the American Statistical Association, 68, 713–715.MathSciNetCrossRefGoogle Scholar
  16. Shuster, E. F. (1975), “Estimating the Distribution Function of a Symmetric Distribution,” Biometrika, 62, 631–635.MathSciNetCrossRefGoogle Scholar
  17. Stephens, M. A. (1991), Tests of Fit for the Cauchy Distribution Based on the Empirical Distribution Function, Report N00014-89-J-1627, Stanford, CA.Google Scholar
  18. Stephens, M. A. and D’Agostino, R. B. (1986), Goodness-of-Fit Techniques, Marcel Decker, New York.zbMATHGoogle Scholar
  19. Yen, V. C., Moore, A. H., and Khatri, D. C. (1997), “Modified Goodness-of-Fit Tests for the Cauchy Distribution,” AFIT/ENC Working Paper, Air Force Institute of Technology.Google Scholar

Copyright information

© Physica-Verlag 2001

Authors and Affiliations

  • Bora H. Onen
    • 1
  • Dennis C. Dietz
    • 1
  • Vincent C. Yen
    • 1
  • Albert H. Moore
    • 1
  1. 1.Air Force Institute of Technology, AFIT/ENWright-Patterson Air Force BaseUSA

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