ESTIMATION OF A MULTIVARIATE BOX-COX TRANSFORMATION TO ELLIPTICAL SYMMETRY VIA

THE EMPIRICAL CHARACTERISTIC FUNCTION

ADOLFO J. QUIROZ^{1}, MIGUEL NAKAMURA^{2} AND FRANCISCO J. PÉREZ^{2}

1Universidad Simón Bolívar, Departamento de Matemáticas, Apartado Postal 89000,

Valle de Sartenejas, Baruta, Caracas, Venezuela 2Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato, Gto. 36000, Mexico

(Received February 6, 1995; revised October 16, 1995)

Abstract. Let X = (X_{1}, X_{2}, …, X_{d})^{t} be a random vector of positive entries, such that for some λ λ λ λ λ λ λ λ = (λ_{1}, λ_{2}, …, λ_{d})^{t}, the vector X^{(λ λ λ λ λ λ λ λ)} defined by X_{i}^{(λ_{i})} = (X_{i}^{λ_{i}} − 1)/λ_{i}, i = 1, …, d is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the λ_{i}’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.