A Minimum Distance Estimation for the Stable Distribution via the Empirical Characteristic Function

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.