The stable distribution is used in various fields because it is a generalization of the normal, Cauchy, Lévy distributions and it has desirable properties such as infinitely divisible distribution. However, conducting the maximum likelihood methods for the stable distributions is very difficult since the probability density function has the complicated form and it is difficult to calculate it. Meanwhile, the characteristic function (cf) of the stable distribution has closed-form expressions. So, the methods of parameter estimation for the stable distribution, based on the cf, which is referred to as the minimum distance estimation (MDE) method has been proposed. However, in the MDE method, we found that the smaller the value of the parameter α, the greatly larger the bias and RMSE of the estimators in this method.
Motivated by the above-mentioned problem, in this article, we propose the new parameter estimation method based on the MDE method, for the stable distribution, that is robust to outliers. To remedy this problem, The key idea of the proposed method is to use a robust minimum distance estimation. Through Monte Carlo simulations, we evaluate the performance of the proposed estimators compared with the existing MDE method in terms of bias and RMSE and show the proposed method performs satisfactorily for all α, even the case of the small α. Furthermore, in an illustrative example, we demonstrate the proposed method outperforms the existing method.
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