Journal of Probability and Statistics

Volume 2017 (2017), Article ID 3483827, 12 pages

https://doi.org/10.1155/2017/3483827

## -Statistic for Multivariate Stable Distributions

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

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. This 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.

#### Abstract

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. The proposed estimator is used to estimate the spectral measure. The 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 fields including physics, economics, finance, insurance, and telecommunications. Different 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 processes. Stable distribution has originated with the work of Lévy [1]. There are a variety of ways to introduce a stable random vector. In the following, two definitions are proposed for a stable random vector; see Samorodnitsky and Taqqu [2].

*Definition 1. *A random vector is said to be stable in if for any positive numbers and there are a positive number and a vector such thatwhere and are independent and identical copies of and .

*Definition 2. *Let . Then is a non-Gaussian -stable random vector in if there exist a finite measure on the unit sphere and a vector such thatwhere for , , , and denotes the sign function. The pair is unique.

The parameter , in Definitions 1 and 2, is called tail index. A random vector is said to be a strictly -stable random vector in if for ; see Samorodnitsky and Taqqu [2]. We note that is strictly -stable, in the sense of Definition 1, if . Throughout we assume that is strictly -stable and . The probability density function of a stable distribution has no closed-form expression and moments with orders greater than or equal to are not finite for the members of this class. The two aforementioned difficulties make statistical inference about the parameters of a stable distribution hard. However, a series of contributions has permitted inference about the parameters of univariate and multivariate stable distributions. For example, in the univariate case, maximum likelihood (ML) estimation was studied first by DuMouchel (1971) and then by Nolan [3]. Although the ML approach leads to an efficient estimate for samples of large size, it involves numerical complexities. A program, called STABLE, uses a cubic spline interpolation of stable densities for this purpose; see Nolan [4]. STABLE estimates all four parameters of a stable distribution for . Sample quantile (SQ) technique is another approach proposed by McCulloch [5]. The results are simple and consistent estimators of all four parameters based on five sample quantiles. The empirical characteristic function (CF) is suggested by Kogon and Williams [6]. The CF and SQ methods work well but are not as efficient as the ML method. As the last approach considered here, -statistics for the tail index and scale parameters of a univariate strictly stable distribution are introduced by Fan [7]. In multivariate case, the focus of interest is the spectral measure estimation. Among them, we refer to Nolan et al. [8], Pivato and Seco [9], Ogata [10], and Mohammadi et al. [11].

The structure of the paper is as follows. In Section 2, new estimators for the tail index and spectral measure of a strictly stable distribution are presented which is an extension of the -statistic proposed by Fan [7] for the univariate case. A comprehensive simulation study is performed in Section 3 to compare the performance of the introduced estimators and the known estimators. Two real data sets are analyzed in this section to illustrate the performance of the proposed method.

#### 2. New Estimators

This section consists of two subsections. Firstly, we propose an estimator for the tail index. Secondly, an estimator for the spectral measure is given.

##### 2.1. Estimation of Tail Index

The main result of this section is given in Theorem 4, which gives -statistic for the inverse of tail index of a strictly stable distribution. We present the main result in the light of Lemma 3 given as follows. The proofs are given in the Appendix.

Lemma 3. *Let be a -dimensional strictly stable random vector. Then, is finite, where denotes the Euclidean norm.*

Theorem 4. *Let be a sequence of observations from a -dimensional strictly stable random vector. Then whereis the -statistic for .*

As it is seen, from Theorem 4, the introduced -statistic is an unbiased estimator for . Hereafter, we write as introduced estimator for . Here, subscript MU indicates that is constructed based on multivariate -statistic defined in Theorem 4. It should be noted that when the true value of is near two, the kernel given in (4) could be less than 0.5. So, is greater than two. In this case, we set .

##### 2.2. Spectral Measure Estimation

We use to estimate an -point discrete approximation to the exact spectral measure of the formwhere is a mass at point in the unit sphere and is an indicator function at point ; for , see Byczkowski et al. [12]. To estimate , we replace Definition 2 for a strictly -dimensional stable random vector withwhere for and for . Definewhere , for . Using (7) and (8), both sides of (6) are connected together through the following linear system:where . Assuming that in (9) is nonsingular, then . Hence, we estimate the vector of the masses aswherein which is -th vector observation in random sample of size .

Due to the standard error of , we have two problems with direct use of (10). Firstly, may be complex, and secondly, its real part may be quite negative. Since and are complex while gamma is constrained to be real (and nonnegative), the Euclidean norm used by McCulloch [13] and Nolan et al. [8] must be replaced with the complex modulus to solve both problems in a novel way. For this, we use the library in the package. In the next section, the estimated spectral measure , based on , is shown by . We note that another estimator of can be constructed by separating both of the real and imaginary parts in the structure of . But simulation results show that constructed estimator gives the same performance.

#### 3. Simulation Study

This section is in three parts. Firstly, we study the performance of the proposed estimator with the known ones for estimating the tail index. Secondly, we compare the performance of the spectral measure estimator developed through the introduced tail index estimator with the known approaches. In the last subsection, we give a real data example to illustrate the efficiency of the proposed estimators.

##### 3.1. Performance Analysis of the Tail Index Estimators

Here, we perform a simulation study to compare the performance of and four other estimators for , including () , () , () , and () . The first three competitors are ML, SQ, and CF estimations for the tail index, respectively. Each of three competitors is obtained as after projecting the -dimensional stable random vector using . Here, is the number of masses, is an arbitrary unit vector, and is the -dimensional stable random vector. It is worth noting that the first three competitors are computed by the help of software after projecting. The fourth estimator, that is, , is the second estimator for tail index proposed by Mohammadi et al. [11]. We compare both the bias and root mean-squared error (RMSE) of estimators for 500 replications of samples of size and of a bivariate stable random vector generated by the method given in Modarres and Nolan [14]. We use two settings for discrete spectral measure with masses, including and . In both cases, masses are concentrated on points = for . In the first case that data are coming from a stable distribution with , we generate from a uniform distribution on the unit sphere . For the second case, we set . Biases and RMSEs for = (0.1 : 0.1 : 0.9, 0.95, 1.05, 1.1 : 0.1 : 1.9, 1.95, 2) are shown in Figures 1 and 2. As Figure 1 shows, when , we observe that is more efficient than for . Also, it works better than in terms of RMSE (for ). Based on Figure 2, when , we observe that is more efficient than other methods when and in the sense of RMSE. Also, when and , is more efficient than , , and with respect to RMSE.