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A class of uniform tests for goodness-of-fit of the multivariate \(L_p\)-norm spherical distributions and the \(l_p\)-norm symmetric distributions

  • Jiajuan LiangEmail author
  • Kai Wang Ng
  • Guoliang Tian
Article
  • 61 Downloads

Abstract

In this paper we employ the conditional probability integral transformation (CPIT) method to transform a d-dimensional sample from two classes of generalized multivariate distributions into a uniform sample in the unit interval \((0,\,1)\) or in the unit hypercube \([0,\,1]^{d-1}\) (\(d\ge 2\)). A class of existing uniform statistics are adopted to test the uniformity of the transformed sample. Monte Carlo studies are carried out to demonstrate the performance of the tests in controlling type I error rates and power against a selected group of alternative distributions. It is concluded that the proposed tests have satisfactory empirical performance and the CPIT method in this paper can serve as a general way to construct goodness-of-fit tests for many generalized multivariate distributions.

Keywords

Goodness-of-fit Monte Carlo study \(L_p\)-norm spherical distribution \(l_p\)-norm symmetric distribution Uniformity 

References

  1. Anderson, T. W. (1993). Nonnormal multivariate distributions: Inference based on elliptically contoured distributions. In C. R. Rao (Ed.), Multivariate analysis: Future directions (pp. 1–24). Amsterdam, London: Elsevier (North Holland Publishing Corporation).Google Scholar
  2. Anderson, T. W., Fang, K. T., Hsu, H. (1986). Maximum likelihood estimates and likelihood ratio criteria for multivariate elliptically contoured distributions. Canadian Journal of Statistics, 14, 55–59.Google Scholar
  3. Fang, K. T., Liang, J. (1999). Testing spherical and elliptical symmetry. In S. Kotz, C. B. Read, & D. L. Banks (Eds.), Encyclopedia of statistical sciences (update) (Vol. 3, pp. 686–691). New York: Wiley.Google Scholar
  4. Fang, K. T., Zhang, Y. (1990). Generalized multivariate analysis. Berlin: Springer.Google Scholar
  5. Fang, K. T., Kotz, S., Ng, K. W. (1990). Symmetric multivariate and related distributions. London and New York: Chapman and Hall.Google Scholar
  6. Gupta, A. K., Kabe, D. G. (1993). Multivariate robust tests for spherical symmetry with applications to multivariate least squares regression. Journal of Applied Statistical Science, 1(2), 159–168.Google Scholar
  7. Gupta, A. K., Song, D. (1997). $L_p$-norm spherical distributions. Journal of Statistical Planning & Inference, 60, 241–260.Google Scholar
  8. Gupta, A. K., Varga, T. (1993). Elliptically contoured models in statistics. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  9. Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299–322.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Huffer, F. W., Park, C. (2007). A test for elliptical symmetry. Journal of Multivariate Analysis, 98, 256–281.Google Scholar
  11. Kariya, T., Eaton, M. L. (1977). Robust tests for spherical symmetry. The Annals of Statistics, 5, 206–215.Google Scholar
  12. Lange, K. L., Little, R. J. A., Taylor, J. M. G. (1989). Robust statistical modeling using the $t$-distribution. Journal of the American Statistical Association, 84, 881–896.Google Scholar
  13. Liang, J., Fang, K. T. (2000). Some applications of Läuter’s technique in tests for spherical symmetry. Biometrical Journal, 42, 923–936.Google Scholar
  14. Liang, J., Ng, K. W. (2008). A method for generating uniformly scattered points on the $L_p$-norm unit sphere and its applications. Metrika, 68, 83–98.Google Scholar
  15. Liang, J., Fang, K. T., Hickernell, F. J., Li, R. (2001). Testing multivariate uniformity and its applications. Mathematics of Computation, 70, 337–355.Google Scholar
  16. Liang, J., Fang, K. T., Hickernell, F. J. (2008). Some necessary uniform tests for spherical symmetry. Annals of the Institute of Statistical Mathematics, 60, 679–696.Google Scholar
  17. Manzottia, A., Pérez, F. J., Quiroz, A. J. (2002). A Statistic for testing the null hypothesis of elliptical symmetry. Journal of Multivariate Analysis, 81, 274–285.Google Scholar
  18. Miller, F. L, Jr., Quesenberry, C. P. (1979). Power studies of tests for uniformity, II. Communications of Statistics-Simulation and Computation B, 8(3), 271–290.Google Scholar
  19. Neyman, J. (1937). “Smooth” test for goodness of fit. Journal of the American Statistical Association, 20, 149–199.zbMATHGoogle Scholar
  20. O’Reilly, F. J., Quesenberry, C. P. (1973). The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests. The Annals of Statistics, 1, 74–83.Google Scholar
  21. Osiewalski, J., Steel, M. F. J. (1993). Robust Bayesian inference in $l_q$-spherical models. Biometrika, 80, 456–460.Google Scholar
  22. Rosenblatt, M. (1952). Remarks on a multivariate transformation. The Annals of Mathematical Statistics, 23, 470–472.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Schott, J. R. (2002). Testing for elliptical symmetry in covariance-matrix-based analysis. Statistics & Probability Letters, 60, 395–404.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Stephens, M. A. (1970). Use of the Kolmogorov Smirnov, Cramér-von Mises and related statistics without extensive tables. Journal of the Royal Statistical Society (Series B), 32, 115–122.zbMATHGoogle Scholar
  25. Watson, G. S. (1962). Goodness-of-fit tests on a circle. II. Biometrika, 49, 57–63.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Yue, X., Ma, C. (1995). Multivariate $l_p$-norm symmetric distributions. Statistics & Probability Letters, 24, 281–288.Google Scholar
  27. Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-$t$ error terms. Journal of the American Statistical Association, 71, 400–405.MathSciNetzbMATHGoogle Scholar
  28. Zhu, L. X. (2003). Conditional tests for elliptical symmetry. Journal of Multivariate Analysis, 84, 284–298.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  1. 1.College of BusinessUniversity of New HavenWest HavenUSA
  2. 2.Department of Statistics and Actuarial ScienceThe University of Hong KongHong KongChina
  3. 3.Department of MathematicsSouthern University of Science and TechnologyNanshan DistrictChina

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