Advertisement

Advances and Challenges in Inferences for Elliptically Contoured t Distributions

  • Brajendra C. SutradharEmail author
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 218)

Abstract

When a multivariate elliptical such as t response is taken from each of n individuals, the inference for the parameters of the t distribution including the location (or regression effects), scale and degrees of freedom (or shape) depends on the assumption whether n multi-dimensional responses are independent or uncorrelated but dependent. In the former case, that is, when responses are independent, the exact sampling theory based inference is extremely complicated, whereas in the later case the derivation of the exact sampling distributions for the standard statistics is manageable but the estimators based on certain standard statistics such as sample covariance matrix may be inconsistent for the respective parameters. In this paper we provide a detailed discussion on the advances and challenges in inferences using uncorrelated but dependent t samples. We then propose a clustered regression model where the multivariate t responses in the cluster are uncorrelated but such clustered responses are taken from a large number of independent individuals. The inference including the consistent estimation of the parameters of this proposed model is also presented.

Keywords

Clustered regression model with uncorrelated t errors Consistent estimation Elliptically contoured distribution Multivariate t and normal as special cases Normality based testing yielding degrees of freedom based power property Regression effects Scale matrix Shape or degrees of freedom parameter 

Notes

Acknowledgements

The author thanks the audience for their comments and suggestions.

References

  1. Amemiya, T.: Advanced Econometrics. Harvard University Press, Cambridge, MA (1985)Google Scholar
  2. Anderson, T.W., Fang, K.T., Hsu, H.: Maximum likelihood estimates and likelihood ratio criteria for multivariate elliptically contoured distributions. Can. J. Stat. 14, 55–59 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chib, S., Tiwari, R.C., Jammalamadaka, S.R.: Bayes prediction in regressions with elliptical errors. J. Econ. 38, 349–360 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chib, S., Osiewalski, J., Steel, M.F.J.: Posterior inference on the degrees of freedom parameter in multivariate-t regression model. Econ. Lett. 37, 391–397 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cornish, E.A.: The multivariate t distribution associated with a set of normal sample deviates. Aust. J. Phys. 7, 531–542 (1954)MathSciNetzbMATHGoogle Scholar
  6. Dunnett, C.W., Sobel, M.: A bivariate generalization of Student’s t-distribution with tables for certain special cases. Biometrika 41, 153–169 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fisher, R.A.: Applications of “Student’s” distribution. Metron 5, 90–104 (1925)zbMATHGoogle Scholar
  8. Fisher, R.A., Healy, M.J.R.: New tables of Behren’s test of significance. J. R. Stat. Soc. B 18, 212–216 (1956)MathSciNetzbMATHGoogle Scholar
  9. Ghosh, B.K.: On the distribution of the difference of two t-variables. J. Am. Stat. Assoc. 70, 463–467 (1975)zbMATHGoogle Scholar
  10. Johnson, N.L., Kotz, S.: Continuous Multivariate Distributions. Wiley, New York (1972)zbMATHGoogle Scholar
  11. Kariya, T.: Robustness of multivariate tests. Ann. Stat. 9, 1267–1275 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kelker, D.: Distribution theory of spherical distribution and a location scale parameter generalization. Sankhya A 32, 419–430 (1970)MathSciNetzbMATHGoogle Scholar
  13. Kotz, S., Nadarajah, S.: Multivariate T-Distributions and Their Applications. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  14. Kubokawa, T., Srivastava, M.S.: Robust improvements in estimation of mean and covariance matrices in elliptically contoured distribution. Technical Report No. 9714. Department of Statistics, University of Toronto (1977)Google Scholar
  15. Mardia, K.V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 47–55 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)CrossRefzbMATHGoogle Scholar
  17. Pearson, K.: Tables of the Incomplete β-Function. University Press, London (1934)Google Scholar
  18. Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. Harvard University Press,Cambridge, MA (1961)zbMATHGoogle Scholar
  19. Student: Probable error of a correlation coefficient. Biometrika 6, 302–310 (1908)Google Scholar
  20. Sutradhar, B.C.: On the characteristic function of multivariate Student t-distribution. Can. J. Stat. 14, 329–337 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sutradhar, B.C.: Testing linear hypothesis with t error variable. Sankhya B 40, 175–180 (1988)MathSciNetzbMATHGoogle Scholar
  22. Sutradhar, B.C.: Discrimination of observations into one of two t populations. Biometrics 46, 827–835 (1990)CrossRefGoogle Scholar
  23. Sutradhar, B.C.: Score test for the covariance matrix of the elliptical t-distribution. J. Multivar. Anal. 46, 1–12 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Sutradhar, B.C.: On cluster regression and factor analysis models with elliptic t errors. In: Anderson, T.W., Fang, K.T., Olkin, I. (eds.) IMS Lecture Notes-Monograph Series, Institute of Mathematical Statistics, Hayward, California, vol. 24, pp. 369–383 (1994)MathSciNetGoogle Scholar
  25. Sutradhar, B.C.: Multivariate t distribution. In: update volume 2 of Encyclopedia of Statistical Sciences. Wiley, New York, pp. 440–448 (1998)Google Scholar
  26. Sutradhar, B.C.: On the consistency of the estimators of the parameters of elliptically contoured distributions. In the special issue of Journal of Statistical Sciences published in honour of Professor M. S. Haq, U. W. O., ed. B.K. Sinha, Indian Statistical Institute, 3, 91–103 (2004)Google Scholar
  27. Sutradhar, B.C.: Dynamic Mixed Models for Familial Longitudinal Data. Springer Series in Statistics, 512 pp. Springer, New York (2011)Google Scholar
  28. Sutradhar, B.C., Ali, M.M.: Estimation of the parameters of a regression model with a multivariate t error variable. Commun. Stat. Theory Methods 15, 429–450 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sutradhar, B.C., Ali, M.M.: A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model. J. Multivar. Anal. 29, 155–162 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Walker, G.A., Saw, J.G.: The distribution of linear combinations of t variables. J. Am. Stat. Assoc. 73, 876–878 (1978)MathSciNetzbMATHGoogle Scholar
  31. Yuen, K.K., Murthy, V.K.: Percentage points of the t-distribution of the t statistic when the parent is Student’s t. Technometrics 16, 495–497 (1974)Google Scholar
  32. Zellner, A.: Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. J. Am. Stat. Assoc. 71, 400–405 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada

Personalised recommendations