Research Article
-Statistic for Multivariate Stable Distributions
Mahdi Teimouri, Saeid Rezakhah, and Adel Mohammadpour
Department of Statistics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic),
424 Hafez Ave., Tehran 15914, Iran
Correspondence should be addressed to Adel Mohammadpour; adel@aut.ac.ir
Received 5 December 2016; Revised 9 February 2017; Accepted 19 February 2017; Published 3 April 2017
Academic Editor: Steve Su
Copyright © 2017 Mahdi Teimouri et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A -statistic for the tail index of a multivariate stable random vector is given as an extension of the univariate case introduced
by Fan (2006). Asymptotic normality and consistency of the proposed -statistic for the tail index are proved theoretically. Te
proposed estimator is used to estimate the spectral measure. Te performance of both introduced tail index and spectral measure
estimators is compared with the known estimators by comprehensive simulations and real datasets.
1. Introduction
In recent years, stable distributions have received extensive
use in a vast number of felds including physics, economics,
fnance, insurance, and telecommunications. Diferent sorts
of data found in applications arise from heavy tailed or
asymmetric distribution, where normal models are clearly
inappropriate. In fact, stable distributions have theoretical
underpinnings to accurately model a wide variety of pro-
cesses. Stable distribution has originated with the work of
L´ evy [1]. Tere are a variety of ways to introduce a stable
random vector. In the following, two defnitions are proposed
for a stable random vector; see Samorodnitsky and Taqqu [2].
Defnition 1. A random vector X = (
1
,...,
)
is said to
be stable in R
if for any positive numbers and there are
a positive number and a vector D ∈ R
such that
X
1
+X
2
=X + D,
(1)
where X
1
and X
2
are independent and identical copies of X
and = (
+
)
1/
.
Defnition 2. Let 0<<2. Ten X is a non-Gaussian -
stable random vector in R
if there exist a fnite measure Γ on
the unit sphere S
={x = (
1
,...,
)
∈ R
|⟨x, x⟩ = 1}
and a vector = (
1
,...,
)
∈ R
such that
X
(t)= log (exp ( ⟨t,X⟩))=
{
{
{
{
{
{
{
−∫
S
|⟨t, s⟩|
[1 − sgn ⟨t, s⟩ tan (
2
)] Γ (s)+⟨t, ⟩, ̸ = 1,
−∫
S
|⟨t, s⟩| [1 + sgn ⟨t, s⟩
2
log |⟨t, s⟩|] Γ (s)+⟨t, ⟩, = 1,
(2)
where ⟨t, s⟩=∑
=1
for t = (
1
,...,
)
, s = (
1
,...,
)
,
2
= −1, and sgn(⋅) denotes the sign function. Te pair (Γ, )
is unique.
Te parameter , in Defnitions 1 and 2, is called tail index.
A random vector X is said to be a strictly -stable random
vector in R
if = 0 for ̸ =1; see Samorodnitsky and
Taqqu [2]. We note that X is strictly -stable, in the sense
of Defnition 1, if D = 0. Troughout we assume that X is
strictly -stable and ̸ =1. Te probability density function
of a stable distribution has no closed-form expression and
moments with orders greater than or equal to are not
Hindawi
Journal of Probability and Statistics
Volume 2017, Article ID 3483827, 12 pages
https://doi.org/10.1155/2017/3483827