Robust and efficient estimation of the shape parameter of alpha-stable distributions

Abstract

In this paper we consider robust and efficient estimators of the shape parameter of symmetric alpha-stable distributions obtained by using the Minimum Density Power Divergence method introduced in Basu, Harris, Hjort and Jones (1998). We established their high asymptotic efficiency and verified these results in simulations. The functionals corresponding to the estimators have bounded influence functions and simulations confirm their robustness when the sample distribution is in a vicinity of the model distribution. The simulations also show that the Minimum Density Power Divergence Estimators (MDPDEs) of the shape parameter of alpha-stable distributions have superior performance over other existing estimators. The high efficiency combined with robustness of the MDPDEs in estimating the shape parameter of alpha-stable distributions make them an attractive alternative to the preceding estimation procedures considered in the literature.