A Method for the Evaluation of Cumulative Probabilities of Bivariate Distributions using the Pearson Family

  • Rudolph S. Parrish
  • Rolf E. Bargmann
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)


A technique is developed for the approximation of bivariate cumulative distribution function values through the use of the bivariate Pearson family when moments of the distribution to be approximated are known. The method utilizes a factorization of the joint density function into the product of a marginal density function and an associated conditional density, permitting the expression of the double integral in a form amenable to the use of specialized Gaussian-type quadrature techniques for numerical evaluation of cumulative probabilities. Such an approach requires moments of truncated Pearson distributions, for which a recurrence relation is presented, and moments of the conditional distributions for which quartic expressions are used. It is shown that results are of high precision when this technique is employed to evaluate cumulative distribution functions that are of the Pearson class.

Key Words

Bivariate distribution Pearson system evaluation of distribution functions quadrature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aiuppa, T.A. (1975). Distributions of linear combinations of quadratic forms (four-moment approximations) with applications. Ph.D. dissertation, Department of Statistics, University of Georgia.Google Scholar
  2. Bargmann, R.E., Halfon, E. (1977). Efficient algorithms for statistical estimation in compartmental analysis: modelling 60Co kinetics in an aquatic microcosm. Ecological Modelling, 3, 211–226.CrossRefGoogle Scholar
  3. Bouver, H., Bargmann, R.E. (1974). Tables of the standardized percentage points of the Pearson system of curves in terms of β1 and β2. Themis Technical Report No. 32, University of Georgia.Google Scholar
  4. Bouver, H., Bargmann, R.E. (1978). Optimaizing the Karl Pearson coefficients in statistical curve fitting. American Statistical Association, Proceedings of the Statistical Computing Section, 314–319Google Scholar
  5. Cohen, A.C. (1951). Estimation of parameters in truncated Pearson frequency distributions. Annals of Mathematical Statistics, 22, 256–265.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Elderton, W. P., Johnson, N. L. (1969). Systems of Frequency Curves. Cambridge University Press, London.zbMATHCrossRefGoogle Scholar
  7. Hamming, R.W. (1973). Numerical Methods for Scientists and Engineers. McGraw-Hill, New York.zbMATHGoogle Scholar
  8. Johnson, N.L. (1949). Bivariate distributions based on simple translation systems. Biometrika, 36, 297–304.MathSciNetGoogle Scholar
  9. Johnson, N.L., Kotz, S. (1970). Distributions in Statistics: Continuous Distributions, Vol. 1, 2. Houghton-Mifflin, Boston.zbMATHGoogle Scholar
  10. Johnson, N.L., Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.zbMATHGoogle Scholar
  11. Lether, E.M. (1978). Application of nonstandard GaussianQuadrature to multiple integrals in statistics. Master’s thesis, Department of Statistics, University of Georgia.Google Scholar
  12. Mardia, K.V. (1970). Families of Bivariate Distributions. Griffin, London.zbMATHGoogle Scholar
  13. Ord, J.K. (1972). Families of Frequency Distributions. Hafner, New York.zbMATHGoogle Scholar
  14. Parrish, R.S. (1978). Evaluation of bivariate distribution functions using the bivariate Fearson family. Ph.D. dissertation, Department of Statistics, University of Georgia.Google Scholar
  15. Pearson, E.S. (1963). Some problems arising in approximating to probability distributions. Biometrika, 50, 95–111.MathSciNetzbMATHGoogle Scholar
  16. van Uven, M.J. (1947, 1948 ). Extension of Pearson’s probability distributions to two variables, I-IV. Nederlandsche Akademie Van Wetenschappen, 50(1947); 1063–1070, 1252–1264; 51(1958), 41–52, 191–196.Google Scholar
  17. White, J.E. (1960). Some contributions to the evaluation of Pearsonian distribution functions. NIH Technical Report No. 1, Virginia Polytechnic Institute.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Rudolph S. Parrish
    • 1
  • Rolf E. Bargmann
    • 2
  1. 1.Computer Sciences Corporation EPA Environmental Research LaboratoryAthensUSA
  2. 2.Department of Statistics University of GeorgiaAthensUSA

Personalised recommendations