Abstract
Let X=(X 1, X 2,..., X d )t be a random vector of positive entries, such that for some λ=(λ1,λ2,...,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.
Similar content being viewed by others
References
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.
Andrews, D. F., Gnanadesikan, R. and Warner, J. L. (1971). Transformations of multivariate data, Biometrics, 27, 825–840.
Arcones, M. A. and Giné, E. (1993). Limit theorems for U-processes, Ann. Probab., 21, 1494–1542.
Assouad, P. (1983). Densité et dimension, Ann. Inst. Fourier Grenoble, 33, 233–282.
Berry, D. A. (1987). Logarithmic transformation in anova, Biometrics, 3, 39–52.
Boos, D. D. (1982). A test for asymmetry associated with a Hodges-Lehmann estimator, J. Amer. Statist. Assoc., 77, 647–651.
Cambanis, S., Huang, S. and Simons, G. (1981). On the theory of elliptically contoured distributions, J. Multivariate Anal., 11, 368–385.
Csörgő, S. (1986). Testing for normality in arbitrary dimension, Ann. Statist., 14, 708–723.
Csörgő, S. and Heathcote, C. R. (1987). Testing for symmetry, Biometrika, 74, 177–184.
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.
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.
Dudley, R. M. (1987). Universal Donsker classes and metric entropy, Ann. Probab., 15, 1306–1326.
Fang, K. T. and Anderson, T. W. (eds.) (1990). Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press, New York.
Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric multivariate and related distributions, Monographs Statist. Appl. Probab., 36, Chapman and Hall, London.
Feuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications, Ann. Statist., 5, 88–97.
Ghosh, S. and Ruymgaart, F. (1992). Applications of empirical characteristic functions in some multivariate problems, Canad. J. Statist., 20(4), 429–440.
Hinkley, D. V. (1977). On quick choice of power transformation, Appl. Statist., 26, 67–68.
Loève, M. M. (1955). Probability Theory: Foundations, Random Sequences, Van Nostrand, New York.
Nakamura, M. and Ruppert, D. (1990). Semi-parametric estimation of symmetrizing transformations with application to the shifted power transformation (unpublished manuscript).
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.
Pollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.
Randles, R. H. and Wolfe, D. A. (1979). Introduction to the Theory of Nonparametric Statistics, Wiley, New York.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.
Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator, Econometrica, 61, 123–137.
Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments, Mathematical Surveys, Number 1, American Mathematical Society, Rhode Island.
Taylor, J. M. G. (1985). Power transformations to symmetry, Biometrika, 72, 145–152.
Velilla, S. (1993). A note on the multivariate Box-Cox transformation to normality, Statist. Probab. Lett., 17, 259–263.
Author information
Authors and Affiliations
Additional information
Adolfo Quiroz and Miguel Nakamura's research was partially supported by CONACYT (Mexico) grants numbers 1858E9219 and 4224E9405, while Dr. Quiroz was visiting Centro de Investigación en Matemáticas at Guanajuato, Mexico.
About this article
Cite this article
Quiroz, A.J., Nakamura, M. & Pérez, F.J. Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function. Ann Inst Stat Math 48, 687–709 (1996). https://doi.org/10.1007/BF00052328
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00052328