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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 [
A random vector
Let
The parameter
The structure of the paper is as follows. In Section
This section consists of two subsections. Firstly, we propose an estimator for the tail index. Secondly, an estimator for the spectral measure is given.
The main result of this section is given in Theorem
Let
Let
As it is seen, from Theorem
We use
Due to the standard error of
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.
Here, we perform a simulation study to compare the performance of
Biases and RMSEs of estimators when data are generated from a strictly stable distribution with discrete spectral measure
Biases and RMSEs of estimators when data are generated from a strictly stable distribution with discrete spectral measure
Here, we compare the performance of the estimator for masses of spectral measure
Independent case:
Symmetric case:
Uniform case:
Triangle case:
Exchangeable case:
RMSEs of
RMSEs of
RMSEs of
RMSEs of
Here, we give two examples. In the first example, adjusted daily logreturn (in percent) for the 30 stocks at the Dow Jones index is collected between January 3, 2000, and December 31, 2004. The logreturn percent of 1247 closing prices has been computed for AXP (American Express Company) and MRK (Merck & Co. Inc.) stocks after multiplying the daily logreturn by 100; see Nolan [
Estimation results after fitting a strictly bivariate
Estimator 














0.396  0.070  0.077  0.433  0.087  0  0.231  0.378  0  0.439  0.078  0 

0.373  0  0  0.463  0.021  0  0.333  0.479  0  0.563  0.100  0 

0.338  0.156  0.162  0.412  0  0  0.490  0.206  0.055  0.425  0.122  0 

0.232  0.216  0.089  0.404  0.012  0  0.480  0.115  0.177  0.254  0.156  0 

0.421  0.129  0.237  0.434  0  0  0.537  0.209  0  0.527  0.089  0 
Estimation results after fitting a strictly bivariate
Estimator 










0.006  1.459  0.498  0  0  0  0  0 

0  0.800  0.257  0  0.026  0.483  0.165  0 

0  1.380  0.508  0  0  0  0  0 

0  1.490  0.532  0  0  0  0  0 

0  0.196  0.199  0  0.009  1.283  0.339  0 
Scatter plot for AXP versus MRK daily logreturn percent,
Scatter plot for cubicroot of Odra and Wisla rivers discharge.
We compare the performance of the introduced
We show that
We rewrite Definition
The authors declare that they have no conflicts of interest.