On estimating the tail index and the spectral measure of multivariate $$\alpha $$ α -stable distributions

Abstract

We propose estimators for the tail index and the spectral measure of multivariate α-stable distributions and derive their asymptotic properties. Simulation studies reveal the appropriateness of the estimators. Applications to financial data are also considered.

U-Statistic for Multivariate Stable Distributions

“…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.…”

“…It is worth noting that the first three competitors are computed by the help of STABLE software after projecting. The fourth estimator, that is,̂M M , 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 = 500 and 5000 of a bivariate stable random vector generated by the method given in Modarres and Nolan [14].…”

“…, ) constructed based on -statistic,̂M U 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)̂M LE-cf ; (2)̂S Q-cf ; (3) CF-cf ; and (4) Mohammadi et al [11] estimator for ,̂M M . For computinĝM LE-cf ,̂S Q-cf , and̂C F-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.…”

“…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].…”

Measures of Dependence for α-Stable Distributed Processes and Its Application to Diagnostics of Local Damage in Presence of Impulsive Noise

“…In the literature one can find several approaches to spectral measure's estimation, starting from methods based on empirical characteristic functions of bivariate vectors [38,39], through spherical harmonic analysis [40] and quantile lines [41], to generalized empirical likelihood method [42]. In this work, for the illustration purposes, we use the method of estimating of the discrete spectral measure proposed by Nolan et al [38], which is based on one-dimensional projections of bivariate vectors and the empirical characteristic functions.…”