Almost efficient estimation of the parameters of a stable distribution using order statistics

This paper presents an order statistics (OS) approach for estimating the unknown parameters of an α stable distribution. The main difficulty in this estimation problem is the lack of a closed form expression for the probability density function (PDF). We suggest to bypass this difficulty by using the fact that the PDF of a sampled quantiles of any random variable is asymptotically normal. We derive maximum likelihood (ML) estimators for the unknown parameters based on the asymptotic statistics of the sampled quantiles and denote them by OSML. The asymptotic performance of the OSML estimates is then analyzed using the appropriate Cramer Rao bounds (OSCRB). These bounds are used to determine the optimal quantiles to be employed in the estimation procedures. Analysis and simulations show that the optimal OSML estimators are almost efficient in the sense that their performance is inversely proportional to the sample size and very close to the Cramer Rao bound (CRB).