A method for simulating non-normal distributions

Abstract

A method of introducing a controlled degree of skew and kurtosis for Monte Carlo studies was derived. The form of such a transformation on normal deviates [XN(0, 1)] isY =a +bX +cX 2 +dX 3. Analytic and empirical validation of the method is demonstrated.

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Reference note

  1. Brown, B. R., & Lathrop, R. L.The effects of violations of assumptions upon certain tests of the product moment correlation coefficient. Paper presented to American Educational Research Association Annual Meeting, February 5, 1971.

References

  1. Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H., & Tukey, J. W.Robust estimation of location: Survey and advances. Princeton: Princeton University Press, 1972.

    Google Scholar 

  2. Brewer, J. K., & Hills, J. R. Skew and Correction for Explicit Univariate Selection. Proceedings of the 76th Annual Convention of the American Psychological Association, 1968, 218–220.

  3. Brewer, J. K., & Hills, J. R. Univariate selection: The effect of correlation, degree of skew, and degree of restriction.Psychometrika, 1969,34, 347–361.

    Google Scholar 

  4. Brown, K. M., & Conte, S. D. The solution of simultaneous non-linear equations.Proceedings of the 22nd National Conference, Association for Computing Machinery. Washington D.C.: Thompson Book Co., 1967, 111–114.

    Google Scholar 

  5. Chambers, J. M., Mallows, C. L., & Stuck, B. W. A method for simulating stable random variables.Journal of the American Statistical Association, 1976,354, 340–344.

    Google Scholar 

  6. Church, A. E. R. On the means and squared standard deviations of small samples from any population.Biometrika, 1926,18, 321–394.

    Google Scholar 

  7. Devlin, S. J., Gnadadesikan, R., & Kettenring, J. R. Robust estimation and outlier detection with correlation coefficients.Biometrika, 1975,62, 531–545.

    Google Scholar 

  8. Donaldson, T. S. Robustness of theF-test to errors of both kinds and the correlation between the numerator and denominator of theF ratio.Journal of the American Statistical Association, 1963,332, 660–676.

    Google Scholar 

  9. Evans, W. D. On the variance of estimates of the standard deviation and variance.Journal of the American Statistical Association, 1951,46, 220–224.

    Google Scholar 

  10. International Mathematical and Statistical Libraries: Computer Subroutine Libraries in Mathematics and Statistics (Vol. I). Houston, Texas: Suite 510, 6200 Hillcroft, 1974.

  11. Gulliksen, H.Theory of mental tests. New York: Wiley, 1950.

    Google Scholar 

  12. Hammersley, J. M., & Handscomb, D. C.Monte carlo methods. London: Metheun & Co. Ltd., 1965.

    Google Scholar 

  13. Kendall, M. C., & Stewart, A.The advanced theory of statistics, (Vol. 1). London: Charles Griffin & Co., Ltd., 1969.

    Google Scholar 

  14. Knuth, D. E.The art of computer programming: Semi-numerical algorithms, (Vol. 2). Reading, Mass.: Addison-Wesley Pub. Co. 1969.

    Google Scholar 

  15. Patnaik, P. B. The non-central chi square andF distributions and their applications.Biometrika, 1949,36, 202–232.

    Google Scholar 

  16. Pearson, E. S., & Please, N. W. Relation between the shape of population distribution of four simple test statistics.Biometrika, 1975,62, 223–241.

    Google Scholar 

  17. Ramberg, J. R. & Schmeiser, B. W. An approximate method for generating asymmetric random variables.Communications of the Association for Computing Machinery, 1974,17, 78–82.

    Google Scholar 

  18. Scheffé, H.The analysis of variance. New York: John Wiley & Sons. 1959.

    Google Scholar 

  19. “Sophister”. Discussion of small samples drawn from an infinite skew population.Biometrika, 1928,20A, 389–423.

    Google Scholar 

  20. Tukey, J. W. A survey of sampling from contaminated distributions. In I. Olkin [Ed.]Contributions to probability and statistics. Stanford, Cal.: Stanford University Press, 1960, 448–485.

    Google Scholar 

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Correspondence to Allen I. Fleishman.

Additional information

This work was done while the author was at the University of Illinois at Champaign-Urbana.

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Fleishman, A.I. A method for simulating non-normal distributions. Psychometrika 43, 521–532 (1978). https://doi.org/10.1007/BF02293811

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Key words

  • computer simulation
  • departures from normality
  • kurtosis
  • Monte Carlo study
  • skew