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

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

Abstract

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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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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