U-Statistic for Multivariate Stable Distributions

Copyright © 2017 Mahdi Teimouri et al.This is an open access article distributed under theCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A U-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 U-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.


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 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 where X 1 and X 2 are independent and identical copies of X and  = (  +   ) 1/ .Definition 2. Let 0 <  < 2. Then X is a non-Gaussian stable random vector in R  if there exist a finite measure Γ on the unit sphere S  = {x = ( 1 , . . .,   )  ∈ R  | ⟨x, x⟩ = 1} and a vector  = ( 1 , . . .,   )  ∈ R  such that where ⟨t, s⟩ = ∑  =1     for t = ( 1 , . . .,   )  , s = ( 1 , . . .,   )  ,  2 = −1, and sgn(⋅) denotes the sign function.The pair (Γ, ) is unique.
The parameter , in Definitions 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 Definition 1, if D = 0. Throughout we assume that X is strictly -stable and  ̸ = 1.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  ≥ 0.4.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.

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.

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 X = ( 1 , . . .,   )  be a -dimensional strictly stable random vector.Then, Var log ‖X‖ is finite, where ‖ ⋅ ‖ denotes the Euclidean norm.Theorem 4. Let x 1 , . . ., x  be a sequence of  observations from a -dimensional strictly stable random vector.Then where is the -statistic for 1/.
As it is seen, from Theorem 4, the introduced -statistic is an unbiased estimator for 1/.Hereafter, we write αMU = 1/  as introduced estimator for .Here, subscript MU indicates that αMU 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, αMU is greater than two.In this case, we set αMU = 2.
2.2.Spectral Measure Estimation.We use αMU to estimate an -point discrete approximation to the exact spectral measure of the form where where where t  = ( 1 , . . .,   )  ∈ S  , for  = 1, . . ., .Using ( 7) and ( 8), both sides of ( 6) are connected together through the following linear system: where  = ( 1 , . . .,   )  .Assuming that Λ in ( 9) is nonsingular, then  = Λ −1 V. Hence, we estimate the vector of the masses as where in which x  is -th vector observation in random sample of size .
Due to the standard error of V, we have two problems with direct use of (10).Firstly, γ may be complex, and secondly, its real part may be quite negative.Since Λ and V 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 nnls(⋅) library in the R package.In the next section, the estimated spectral measure γ, based on αMU , is shown by γMU .We note that another estimator of  can be constructed by separating both of the real and imaginary parts in the structure of V.But simulation results show that constructed estimator gives the same performance.

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.

Performance Analysis of the Spectral Measure Estimators.
Here, we compare the performance of the estimator for masses of spectral measure  = ( 1 , . . .,   )  constructed based on -statistic, γMU with the other four known estimators for the spectral measure.The competitors are three types of estimators for  based on empirical characteristic function method: (1) γMLE-cf ; (2) γSQ-cf ; (3) γCF-cf ; and (4) Mohammadi et al. [11] estimator for , γMM .For computing γMLE-cf , γSQ-cf , and γCF-cf , we use command mvstable.fit(x,nspectral, method1d, method2d, param) in the STABLE program, where x is data vector, nspectral is number of spectral measure masses, method1d is the method to use for estimating parameters of univariate stable distribution, that is, MLE, SQ, and CF (corresponding codes in STABLE are 1, 2, and 3, respectively), method2d is the method to use for estimating parameters of bivariate stable distribution (we set method2d = 2 which corresponds to empirical characteristic function approach, cf), and param refers to kind of parameterization.Here, we set param = 1 since we are using the characteristic function in (2).More information about the first three competitors is given in Robust Analysis Inc. [15].The estimators γMLE-cf , γSQ-cf , γCF-cf , and γMM are obtained by substituting αML , αSQ , αCF , and αMM into (7) and then solving linear system (10), respectively.Comparisons are based on the RMSE of γ , for  = 1, . . ., , which is defined as √1/ ∑  =1 (γ  −   ) 2 , where  is the number of iterations and γ is the estimation of th component of γ at th iteration.We consider five scenarios for the structure of discrete spectral measure as follows.

Real Data
Table 1 shows the results for modelling data through five methods.We note that estimated tail indices are αMU = 1.581, αMM = 1.734, αML-cf = 1.618, αSQ-cf = 1.493, and αCF-cf = 1.723.As it is seen, estimated tail indices through estimators αMU and αML-cf are closer together than the other estimators.In the second example, we focus on the cubic-root of the monthly average of river discharge.We choose discharge of the Odra and Wisla rivers in Poland during 1901 to 1986 (raw data are in m 3 /s.They are available at https://nelson.wisc.edu/sage/data-and-models/riverdata/).The scatter plot for cubic-root of Odra river discharge versus cubic-root of Wisla river discharge is shown in Figure 8.

Conclusion
We compare the performance of the introduced -statistic for the tail index with the well-known methods, including maximum likelihood, empirical characteristic function, sample quantile, and that introduced in Mohammadi et al. [11] through a simulation study.In the sense of root meansquared error, it is proved that proposed tail index estimator always outperforms Mohammadi et al. [11] and SQ methods when  ≤ 1.4.This is while ML and CF methods show better performance than the proposed estimator for large , say  > 1.4 in terms of root mean-squared error.Simulation studies for estimating the discrete spectral measure  under five scenarios prove that estimator of  based on introduced -statistic shows, in terms of root mean-squared error, better performance than Mohammadi et al. [11] estimator.Analysis of two sets of real data reveals that estimator of the tail index and  based on -statistic shows expedient performance.As some possible future works, firstly, we aim to introduce a -statistic for the case of a nonzero location parameter.Secondly, we look for methodology possibly based on a statistic, to estimate tail, masses, and location parameters simultaneously.Finally, recalling that the approach employed in this work is based on characteristic function, the discrete spectral measure using αMU can be estimated through projection approach.where we adopt this convention that ∑ 0 =1   = 0. Let (, , ,  = 0) stands for a univariate strictly stable random variable with tail index , scale parameter , and skewness parameter .It is well known that if X = ( 1 , . . .,   )  is an -stable random vector, then any linear combination of its components such as ⟨b  , X⟩ = ∑  =1     , for b  = (

Figure 8 :
Figure 8: Scatter plot for cubic-root of Odra and Wisla rivers discharge.

Table 1 :
RMSEs of γ under different scenarios when  = 1.75.We use the following symbol scheme: ⬦ for γUM , I for γMM , + for γML-cf , × for γSQ-cf , and  for γCF-cf .Estimation results after fitting a strictly bivariate -stable distribution to AXP and MRK stocks data.

Table 2 :
Estimation results after fitting a strictly bivariate -stable distribution to Odra and Wisla discharge data.