Abstract
While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly to their limiting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional integrals that are not easily evaluated in numerical computation. In this paper, we develop some necessary tests for spherical symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and have simple limiting distributions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and power.
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References
Anderson T.W. (1993). Nonnormal multivariate distributions: Inference based on elliptically contoured distributions. In: Rao C.R. (eds). Multivariate analysis: Future directions. Elsevier, Amsterdam, pp. 1–24
Baringhaus L. (1991). Testing for spherical symmetry of a multivariate distribution. The Annals of Statistics, 19, 899–917
Fang H., Fang K.-T., Kotz S. (2002). The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis, 82, 1–16
Fang K.-T., Fang B.-Q. (1988). Some families of multivariate symmetric distributions related to exponential distribution. Journal of Multivariate Analysis, 24, 109–122
Fang K.-T., Fang B.-Q. (1989). A characterization of multivariate l 1-norm symmetric distribution. Statistics and Probability Letters, 7, 297–299
Fang K.-T., Kotz S., Ng K.W. (1990). Symmetric multivariate and related distributions. London, Chapman and Hall
Fang K.-T., Liang J. (1999). Testing spherical and elliptical symmetry. In: Kotz S., Read C.B., Banks D.L. (eds). Encyclopedia of statistical sciences (Update). (Vol. 3). New York, Wiley, pp. 686–691
Fang K.-T., Wang Y. (1994). Number-theoretic methods in statistics. London, Chapman and Hall
Fang K.-T., Zhu L.-X., Bentler P.M. (1993). A necessary test for sphericity of a high-dimensional distribution. Journal of Multivariate Analysis, 44, 34–55
Fujikoshi Y. (1997). An asymptotic expansion for the distribution for the distribution of Hotelling T 2-statistic. Journal of Multivariate Analysis, 61, 187–193
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 and Inference, 60, 241–260
Hickernell F.J. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299–322
Kariya T., Eaton M.L. (1977). Robust tests for spherical symmetry. The Annals of Statistics, 5, 206–215
Kariya T., Sinha B.K. (1989). Robustness of statistical tests. New York, Academic Press
Koltchinskii V.I., Li L. (1998). Testing for spherical symmetry of a multivariate distribution. Journal of Multivariate Analysis, 65, 228–244
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
Li R., Fang K.-T., Zhu L.-X. (1997). Some Q-Q probability plots to test spherical and elliptical symmetry. Journal of Computational and Graphical Statistics, 6, 435–450
Liang J., Fang K.-T. (2000). Some applications of Läuter’s technique in tests for spherical symmetry. Biometrical Journal, 42, 923–936
Liang J., Fang K.-T., Hickernell F.J., Li R. (2001). Testing multivariate uniformity and its applications. Mathematics of Computation, 70, 337–355
Miller F.L. Jr., Quesenberry C.P. (1979). Power studies of tests for uniformity, II. Communications in Statistics –Simulation and Computation, B8(3): 271–290
Neyman J. (1937). “Smooth” test for goodness of fit. Journal of the American Statistical Association, 20, 149–199
Osiewalski J., Stel M.F.J. (1993). Robust Bayesian inference on l q -spherical models. Biometrika, 80, 456–460
Quesenberry C.P., Miller F.L., Jr. (1977). Power studies of some tests for uniformity. Journal of Statistical Computation and Simulation, 5, 169–191
Rosenblatt M. (1952). Remarks on a multivariate transformation. The Annals of Mathematical Statistics, 23, 470–472
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
Tashiro D. (1977). On methods for generating uniform points on the surface of a sphere. The Annals of the Institute of Statistical Mathematics, 29, 295–300
Wakaki H. (1994). Discriminant analysis under elliptical distributions. Hiroshima Mathematical Journal, 24, 257–298
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 and 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., Fang, K.-T., Zhang, J.-T. (1995). A projection NT-type test for spherical symmetry of a multivariate distribution. New trends in probability and statistics (Vol. 3, pp. 109–122). Utrecht, The Netherland, Tokyo, Japan: VSP and Uilnius, Lithuania: TEV.
Zhu L.-X., Fang K.-T., Bhatti M.I., Bentler P.M. (1995). Testing sphericity of a high-dimensional distribution based on bootstrap approximation. Pakistan Journal of Statistics, 11(1): 49–65
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Liang, J., Fang, KT. & Hickernell, F.J. Some necessary uniform tests for spherical symmetry. Ann Inst Stat Math 60, 679–696 (2008). https://doi.org/10.1007/s10463-007-0121-9
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DOI: https://doi.org/10.1007/s10463-007-0121-9