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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
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.
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.
Fang, K. T., Zhang, Y. (1990). Generalized multivariate analysis. Berlin: Springer.
Fang, K. T., Kotz, S., Ng, K. W. (1990). Symmetric multivariate and related distributions. London and New York: Chapman and Hall.
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.
Gupta, A. K., Song, D. (1997). $L_p$-norm spherical distributions. Journal of Statistical Planning & Inference, 60, 241–260.
Gupta, A. K., Varga, T. (1993). Elliptically contoured models in statistics. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299–322.
Huffer, F. W., Park, C. (2007). A test for elliptical symmetry. Journal of Multivariate Analysis, 98, 256–281.
Kariya, T., Eaton, M. L. (1977). Robust tests for spherical symmetry. The Annals of Statistics, 5, 206–215.
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.
Liang, J., Fang, K. T. (2000). Some applications of Läuter’s technique in tests for spherical symmetry. Biometrical Journal, 42, 923–936.
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.
Liang, J., Fang, K. T., Hickernell, F. J., Li, R. (2001). Testing multivariate uniformity and its applications. Mathematics of Computation, 70, 337–355.
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.
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.
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.
Neyman, J. (1937). “Smooth” test for goodness of fit. Journal of the American Statistical Association, 20, 149–199.
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.
Osiewalski, J., Steel, M. F. J. (1993). Robust Bayesian inference in $l_q$-spherical models. Biometrika, 80, 456–460.
Rosenblatt, M. (1952). Remarks on a multivariate transformation. The Annals of Mathematical Statistics, 23, 470–472.
Schott, J. R. (2002). Testing for elliptical symmetry in covariance-matrix-based analysis. Statistics & Probability Letters, 60, 395–404.
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.
Watson, G. S. (1962). Goodness-of-fit tests on a circle. II. Biometrika, 49, 57–63.
Yue, X., Ma, C. (1995). Multivariate $l_p$-norm symmetric distributions. Statistics & Probability Letters, 24, 281–288.
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.
Zhu, L. X. (2003). Conditional tests for elliptical symmetry. Journal of Multivariate Analysis, 84, 284–298.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported in part by the University of Hong Kong Research Grant and University of New Haven Research Scholar Grant.
About this article
Cite this article
Liang, J., Ng, K.W. & Tian, G. A class of uniform tests for goodness-of-fit of the multivariate \(L_p\)-norm spherical distributions and the \(l_p\)-norm symmetric distributions. Ann Inst Stat Math 71, 137–162 (2019). https://doi.org/10.1007/s10463-017-0630-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-017-0630-0