*Article* **An Universal, Simple, Circular Statistics-Based Estimator of** *α* **for Symmetric Stable Family**

**Ashis SenGupta 1,2,3 and Moumita Roy 4,\***


Received: 15 October 2019; Accepted: 18 November 2019 ; Published: 23 November 2019

**Abstract:** The aim of this article is to obtain a simple and efficient estimator of the index parameter of symmetric stable distribution that holds universally, i.e., over the entire range of the parameter. We appeal to directional statistics on the classical result on wrapping of a distribution in obtaining the wrapped stable family of distributions. The performance of the estimator obtained is better than the existing estimators in the literature in terms of both consistency and efficiency. The estimator is applied to model some real life financial datasets. A mixture of normal and Cauchy distributions is compared with the stable family of distributions when the estimate of the parameter *α* lies between 1 and 2. A similar approach can be adopted when *α* (or its estimate) belongs to (0.5,1). In this case, one may compare with a mixture of Laplace and Cauchy distributions. A new measure of goodness of fit is proposed for the above family of distributions.

**Keywords:** Index parameter; estimation; wrapped stable; Hill estimator; characteristic function-based estimator; asymptotic; efficiency
