Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function

  • Adolfo J. Quiroz
  • Miguel Nakamura
  • Francisco J. Pérez


Let X=(X1, X2,..., X d ) t be a random vector of positive entries, such that for some λ=(λ12,...,λ d ) t , the vector X(λ) defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\[X_i^{(\lambda _{i)} } = ({{X_i^{\lambda _i } - 1)} \mathord{\left/ {\vphantom {{X_i^{\lambda _i } - 1)} {\lambda _{i,} i = 1 \ldots ,d}}} \right. \kern-\nulldelimiterspace} {\lambda _{i,} i = 1 \ldots ,d}}\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the λi's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.

Key words and phrases

Elliptically contoured distributions empirical characteristic function Box-Cox transformations 


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  1. Andrews, D. F. and Herzberg, A. M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker, Springer, New York.Google Scholar
  2. Andrews, D. F., Gnanadesikan, R. and Warner, J. L. (1971). Transformations of multivariate data, Biometrics, 27, 825–840.Google Scholar
  3. Arcones, M. A. and Giné, E. (1993). Limit theorems for U-processes, Ann. Probab., 21, 1494–1542.Google Scholar
  4. Assouad, P. (1983). Densité et dimension, Ann. Inst. Fourier Grenoble, 33, 233–282.Google Scholar
  5. Berry, D. A. (1987). Logarithmic transformation in anova, Biometrics, 3, 39–52.Google Scholar
  6. Boos, D. D. (1982). A test for asymmetry associated with a Hodges-Lehmann estimator, J. Amer. Statist. Assoc., 77, 647–651.Google Scholar
  7. Cambanis, S., Huang, S. and Simons, G. (1981). On the theory of elliptically contoured distributions, J. Multivariate Anal., 11, 368–385.Google Scholar
  8. Csörgő, S. (1986). Testing for normality in arbitrary dimension, Ann. Statist., 14, 708–723.Google Scholar
  9. Csörgő, S. and Heathcote, C. R. (1987). Testing for symmetry, Biometrika, 74, 177–184.Google Scholar
  10. Devlin, S. J., Gnanadesikan, R. and Kettenring, J. R. (1976). Some multivariate applications of elliptical distributions, Essays in Probability and Statistics (eds. S. Ideka, T. Hayakawa, H. Hudimoto, M. Okamoto, M. Siotani and S. Yamaoto), 365–395, Shink Tsusho Co., Ltd. Tokyo.Google Scholar
  11. Dudley, R. M. (1984). A course on empirical processes, Ecole d'Eté de Probabilités de Saint-Flour, XII-1982, Lecture Notes in Math., 1097, 1–142. Springer, New York.Google Scholar
  12. Dudley, R. M. (1987). Universal Donsker classes and metric entropy, Ann. Probab., 15, 1306–1326.Google Scholar
  13. Fang, K. T. and Anderson, T. W. (eds.) (1990). Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press, New York.Google Scholar
  14. Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric multivariate and related distributions, Monographs Statist. Appl. Probab., 36, Chapman and Hall, London.Google Scholar
  15. Feuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications, Ann. Statist., 5, 88–97.Google Scholar
  16. Ghosh, S. and Ruymgaart, F. (1992). Applications of empirical characteristic functions in some multivariate problems, Canad. J. Statist., 20(4), 429–440.Google Scholar
  17. Hinkley, D. V. (1977). On quick choice of power transformation, Appl. Statist., 26, 67–68.Google Scholar
  18. Loève, M. M. (1955). Probability Theory: Foundations, Random Sequences, Van Nostrand, New York.Google Scholar
  19. Nakamura, M. and Ruppert, D. (1990). Semi-parametric estimation of symmetrizing transformations with application to the shifted power transformation (unpublished manuscript).Google Scholar
  20. Nelson, C. H., Cox, D. D. and Ndjuenga, J. (1989). Mean variance portfolio choice: a test for elliptical symmetry, Tech. Report, No. 41, Department of Statistics, University of Illinois.Google Scholar
  21. Pollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.Google Scholar
  22. Randles, R. H. and Wolfe, D. A. (1979). Introduction to the Theory of Nonparametric Statistics, Wiley, New York.Google Scholar
  23. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.Google Scholar
  24. Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator, Econometrica, 61, 123–137.Google Scholar
  25. Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments, Mathematical Surveys, Number 1, American Mathematical Society, Rhode Island.Google Scholar
  26. Taylor, J. M. G. (1985). Power transformations to symmetry, Biometrika, 72, 145–152.Google Scholar
  27. Velilla, S. (1993). A note on the multivariate Box-Cox transformation to normality, Statist. Probab. Lett., 17, 259–263.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1996

Authors and Affiliations

  • Adolfo J. Quiroz
    • 1
  • Miguel Nakamura
    • 2
  • Francisco J. Pérez
    • 2
  1. 1.Departmento de MatemáticasUniversidad Simón BolívarCaracasVenezuela
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMexico

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