New Results on Truncated Elliptical Distributions
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New Results on Truncated Elliptical Distributions Raúl Alejandro Morán-Vásquez1
· Silvia L. P. Ferrari2
Received: 5 November 2018 / Revised: 21 January 2019 / Accepted: 9 July 2019 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions. Two members of this class are the multivariate truncated normal and multivariate truncated t distributions. We derive statistical properties of the truncated elliptical distributions. Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions. Keywords Elliptical distribution · Log-concavity · Multivariate power exponential distribution · Multivariate slash distribution · Truncated distribution Mathematics Subject Classification 60E05 · 62E15 · 65C05
1 Introduction The class of elliptical distributions (see Fang et al. [5]) is of central relevance in applied and theoretical statistics, in particular in multivariate regression analysis. This class comprises a wide variety of distributions for modeling multivariate data, including those with heavier-than-normal or slighter-than-normal tail behavior. Some elliptical distributions are multivariate normal, t, power exponential, slash, scale mixture of normal distributions, among others. A generalization of this class is constructed by truncation: it is derived as the conditional distribution of a random vector, with dimension p say, having an elliptical distribution given that it belongs to a subset of
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Raúl Alejandro Morán-Vásquez [email protected] Silvia L. P. Ferrari [email protected]
1
Institute of Mathematics, University of Antioquia, Calle 67 No. 53-108, Medellín, Colombia
2
Department of Statistics, University of São Paulo, Rua do Matão 1010, São Paulo 05508-090, Brazil
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R. A. Morán-Vásquez, S. L. P. Ferrari
R p . Some works dealing with truncated elliptical distributions are found in MoránVásquez and Ferrari [16], Kim [15] and Arellano-Valle et al. [2]. The class of truncated elliptical distributions has as special members the multivariate truncated normal and multivariate truncated t distributions. Some studies regarding to these two distributions can be found in Kan and Robotti [14], Arismendi [1], Ho et al. [11], Nadarajah [17], Horrace [12,13], Geweke [7], Tallis [18–20] and Birnbaum and Meyer [3]. However, other distributions, such as the multivariate truncated slash and multivariate truncated power exponential distributions, have not been studied. We derive several properties of the truncated elliptical distributions, thus generalizing some results previously shown for the multivariate truncated normal and multivariate truncated t distributions only. Some of our results consider arbitrary truncation subsets of R p ; others are valid only for p
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