Exact tests and confidence sets for the tail coefficient of a-stable distributions

In this paper, using the Monte Carlo (MC) method we propose an estimation and (at the same time) a test procedure for the stability parameter of a-stable distributions. One powerful advantage of the MC method is that it provides an exact significance level for finite samples, whose distribution can be far different from that of asymptotic samples on which the level of confidence interval for estimates is usually based. Statistical theory for the MC method is given. A simulation study compares the efficiency of our estimate with the Hill estimate (Hill, 1975). Construction of significance level based on the MC method is exploited and the corresponding power function is also studied. An empirical application demonstrates an easy implementation of our estimation and test procedure. It turns out that our estimate can improve the efficiency of any estimator for a in terms of mean square error.