On comparison of the well - known estimators for the stable index in the cy - stable distributions and distributions attracted in their domain

On comparison of the well - known estimators for the stable index in the cy - stable distributions and distributions attracted in their domain

The most important of tail estimators for the stable index of a heavy tailed distributions is the Hills estimator which is often applied to stable distributions with index (; 4 2. and distributions in their domain of attraction. However, for stable distributions with index close to 2. the behavior is not satisfactory. Under the same assumption on the distributions. Press (1972) and Zolotarev (1986) constructed different estimators for stable index based on characteristic function of stable random variables, separately. De Haan and Pereira (1999) developed a further estimator based on the order statistics and described some of the disadvantages of Hills estimator. Fan (2004) establish a new estimator with U-statistics structure for the tail index of heavy-tailed distributions. In this paper, we study these five estimators. and then compare all of them in the sense of the mean square error criteria. The simulation study conducted to evaluate the performances of these estimators in the sense that, whether the distribution of the parent population is Cy-stable. Pressʹs estimator performs better than Zolotarevʹs estimator. Zolota estimator performs better than Fans estimator. Fans estimator performs better than Hills estimator and finally....