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A Kotz-Type Distribution for Multivariate Statistical Inference

  • Dayanand N. Naik
  • Kusaya Plungpongpun
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

In this chapter, we consider a Kotz-type distribution (of a p-variate random vector X) which has fatter tail regions than that of multivariate normal distribution, and its probability density function (pdf) is given by
$$ f(x,\mu ,\Sigma ) = c_p \left| \Sigma \right|^{ - \tfrac{1} {2}} \exp \{ - [(x - \mu )'\Sigma ^{ - 1} (x - \mu )]^{\tfrac{1} {2}} \} , $$
where μ∈ℜp, Σ is a positive definite matrix and \( c_p = \tfrac{{\Gamma (\tfrac{p} {2})}} {{2\pi ^{\tfrac{p} {2}} \Gamma (p)}} \). We review various characteristics and provide a simulation algorithm to simulate samples from this distribution. Estimation of the parameters using the maximum likelihood method is discussed. An interesting fact is that the maximum likelihood estimators under this distribution are the generalized spatial median (GSM) estimators as defined by (1988). Using the asymptotic distribution of the estimates, statistical inferences on the parameters of the distribution are illustrated with an example.

Keywords and phrases

Generalized spatial median Kotz-type distribution simulation algorithm simultaneous confidence intervals 

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References

  1. 1.
    Baringhaus, L., and Henze, N. (1992). Limit distributions for Mardia’s measure of multivariate skewness, Annals of Statistics, 20, 1889–1902.zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brown, B. M. (1983). Statistical use of the spatial median, Journal of the Royal Statistical Society, Series B, 45, 25–30.zbMATHGoogle Scholar
  3. 3.
    Devroye, L. (1986). Non-Uniform Random Variate Generation, Springer-Verlag, New York.zbMATHGoogle Scholar
  4. 4.
    Ducharme, G. R., and Milasevic, P. (1987). Spatial median and directional data. Biometrika, 74, 212–215.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fang, K. T., and Anderson, T. W. (1990). Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press, New York.zbMATHGoogle Scholar
  6. 6.
    Fang, K. T. and Kotz, S., and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions Chapman and Hall, London.zbMATHGoogle Scholar
  7. 7.
    Gómez, E. Gómez-Villegas, M. A., and Marín, J.M. (1998). A multivariate generalization of the power exponential family of distributions, Communications in Statistics—Theory and Methods, 27, 589–600.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gower, J. S. (1974). The mediancentre, Applied Statistics, 23, 466–470.CrossRefGoogle Scholar
  9. 9.
    Haldane, J. B. S. (1948). Note on the median of a multivariate distribution, Biometrika, 35, 414–415.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Henze, N. (1994). On Mardia’s kurtosis test for multivariate normality, Communications in Statistics—Theory and Methods, 23, 1031–1045.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions, In Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 221–233, University of California Press, Berkeley, CA.Google Scholar
  12. 12.
    Huber, P. J. (1981). Robust Statistics, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  13. 13.
    Johnson, R. A., and Wichern, D. W. (1998). Applied Multivariate Statistical Analysis, Prentice Hall, New Jersey.Google Scholar
  14. 14.
    Kano, Y. (1994). Consistency property of elliptical probability density functions, Journal of Multivariate Analysis, 51, 139–147.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kariya, T. K., and Sinha, B. K. (1989). Robustness of Statistical Tests, Academic Press, San Diego, CA.zbMATHGoogle Scholar
  16. 16.
    Khattree, R., and Naik, D. N. (1999). Applied Multivariate Statistics with SAS Software, John Wiley_& Sons and SAS Institute, New York.Google Scholar
  17. 17.
    Kotz, S. (1975). Multivariate distributions at a cross-road, In Statistical Distributions in Scientific Work (Eds., G. P. Patil, S. Kotz, and J. K. Ord), pp. 247–270, D. Reidel, The Netherlands.Google Scholar
  18. 18.
    Kotz, S., Kozubowski, T. J., and Podgórski, K. (2001). The Laplace Distribution and Generalizations, Birkhäuser, Boston.zbMATHGoogle Scholar
  19. 19.
    Koutras, M. (1986). On the generalized noncentral chi-squared distribution induced by an elliptical gamma law, Biometrika, 73, 528–532.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Lange, K. L., Little, R. J. A., and Taylor, J. M. G. (1989). Robust statistical modeling using the t distribution, Journal of the American Statistical Association, 84, 881–896.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lindsey, J. K. (1999). Multivariate elliptically contoured distributions for repeated measurements, Biometrics, 55, 1277–1280.zbMATHCrossRefGoogle Scholar
  22. 22.
    Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519–530.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  24. 24.
    Nadarajah, S. (2003). The Kotz-type distribution with applications, Statistics, 37, 341–358.zbMATHMathSciNetGoogle Scholar
  25. 25.
    Naik, D. N. (1993). Multivariate medians: A review In Probability and Statistics (Eds., S. K. Basu and B. K. Sinha), pp. 80–90, Narosa Publishing House, New Delhi.Google Scholar
  26. 26.
    Naik, D., Khattree, R., and Shults, J. (2002). A note on likelihood based inference for AR(1) and MA(1) processes under certain robust alternatives to multivariate normality, Journal of Statistical Theory and Applications, 1, 57–62.MathSciNetGoogle Scholar
  27. 27.
    Naik, D. N., and Patwardhan, G. R. (1991). A Note on testing for correlation in a certain bivariate distribution, Journal of Quantitative Economics, 7, 295–302.Google Scholar
  28. 28.
    Plungpongpun, K. (2003). Analysis of multivariate data using Kotz type distribution, Ph.D. Thesis, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA.Google Scholar
  29. 29.
    Rao, C. R. (1988). Methodology based on the L 1-norm in statistical inference, Sankhyā, Series A 50, 289–313.zbMATHGoogle Scholar
  30. 30.
    Simoni, S. de (1968) Su una estensione dello schema delle curve normali di ordina r alle variabili doppie Statistica (Bologna), 28, 151–170.Google Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • Dayanand N. Naik
    • 1
    • 2
  • Kusaya Plungpongpun
    • 1
    • 2
  1. 1.Old Dominion UniversityNorfolkUSA
  2. 2.Silpakorn UniversityBangkokThailand

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