A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property
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A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property Yeshunying Wang1 · Chuancun Yin1 Received: 21 June 2019 / Revised: 17 April 2020 / Accepted: 27 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce a new class of multivariate elliptically symmetric distributions including elliptically symmetric logistic distributions and Kotz type distributions. We investigate the various probabilistic properties including marginal distributions, conditional distributions, linear transformations, characteristic functions and dependence measure in the perspective of the inconsistency property. In addition, we provide a real data example to show that the new distributions have reasonable flexibility. Keywords Elliptically contoured distribution · Elliptically symmetric logistic distribution · Kotz type distribution · Inconsistency property · Generalized Hurwitz-Lerch zeta function Mathematics Subject Classification (2010) 60E · 62F
1 Introduction The multivariate normal distribution has been widely used in theory and practice because of its tractable statistical features. However, the light tail of the normal distribution can not fit some practical situation well. The elliptically contoured distributions (elliptical distributions), a new family of distributions with similar convenient properties, overcomes the shortcomings of the normal distributions. An n-dimension random vector X is said to have a multivariate elliptical distribution, written as X ∼ Elln (μ, , φ) if its characteristic function can be expressed as ψX (t) = exp(itT μ)φ(tT t), where μ is an n-dimension column vector, is an n × n positive semi-definite matrix, φ(·) is called characteristic generator. If X has a probability density function (pdf) f (x), then Cn f (x) = √ gn (x − μ)T −1 (x − μ) , || Chuancun Yin
[email protected] 1
School of Statistics, Qufu Normal University, Shandong 273165, China
Methodology and Computing in Applied Probability
where Cn is the normalizing constant and gn (·) is called density generator (d.g.). The stochastic representation of X is given by X = μ + RAT U(n) ,
(1.1)
is uniformly distributed on the unit where A is a square matrix such that A A = , sphere surface in Rn , R ≥ 0 is independent of U(n) and has the pdf given by T
fR (v) = ∞ 0
1 t n−1 g(t 2 )dt
U(n)
v n−1 g(v 2 ), v ≥ 0.
(1.2)
Many members of the elliptical distributions such as the multivariate normal distributions and student-t distributions, have been systematic studied. See the books and papers of Cambanis et al. (1981), Fang et al. (1990), Kotz and Ostrovskii (1994), Liang and Bentler (1998) and Nadarajah (2003). Nevertheless, research work on the multivariate symmetric logistic distribution is far less than other members. The elliptically symmetric logistic distribution with density 1
f (x) =
( n2 )||− 2 exp(−(x − μ)T −1 (x − μ)) , x ∈ Rn , n ∞ n exp(−u) π 2 0 u 2 (1+exp(−u))2 du [1 + exp(−(x − μ)T −1 (x − μ))]2
was introduced by Jensen (
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