Estimation of the stability factor of alpha-stable laws using fractional moments method
Vadim Shergin · Eastern-European Journal of Enterprise Technologies · 2013
The problem of estimating the stability factor of alpha-stable distributions was considered. Such distributions are widely used in models of stochastic processes, describing a wide class of processes and phenomena.The analysis of existing methods of the estimation of parameters of stable distributions was carried out. It was noted that stable distributions do not have moments of the second and higher orders. This makes it impossible to use such classical statistical method for the estimation of parameters as the method of moments.The use of the method of fractional moments for the estimation of parameters of stable distributions is proposed in the paper. The mathematical basis of the method of fractional moments is the theory of Mellin transforms.The estimates of the required factor were obtained in the exact and approximate forms. The consistency and asymptotic unbiasedness of these estimates were proved, and their asymptotic variance was calculated.The values of the moments, which minimize the asymptotic variance of estimates of the stability factor, were found. These values depend on the value of the estimated stability factor.The numerical modeling, which confirmed the obtained results, was conducted