U-Statistic for Multivariate Stable Distributions

Mahdi Teimouri,Adel Mohammadpour,Saeid Rezakhah

doi:10.1155/2017/3483827

Mahdi Teimouri, Adel Mohammadpour + Show 1 more

Open Access

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

Copy DOI

Abstract

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.

Highlights

In recent years, stable distributions have received extensive use in a vast number of fields including physics, economics, finance, insurance, and telecommunications
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, Sample quantile (SQ), and characteristic function (CF), method2d is the method to use for estimating parameters of bivariate stable distribution, and param refers to kind of parameterization
We compare the performance of the introduced U-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

Summary

Introduction

Stable distributions have received extensive use in a vast number of fields including physics, economics, finance, insurance, and telecommunications. There are a variety of ways to introduce a stable random vector. Two definitions are proposed for a stable random vector; see Samorodnitsky and Taqqu [2]. The parameter α, in Definitions 1 and 2, is called tail index. A series of contributions has permitted inference about the parameters of univariate and multivariate stable distributions. As the last approach considered here, U-statistics for the tail index and scale parameters of a univariate strictly stable distribution are introduced by Fan [7]. Two real data sets are analyzed to illustrate the performance of the proposed method

New Estimators

Simulation Study

Conclusion