The distributions for highly volatile financial time-series data are playing an increasingly important role in current financial scenarios and signal analyses. An important characteristic of such a probability distribution is its tail behaviour, determined through its tail thickness. This can be achieved by estimating the index parameter of the corresponding distribution. The normal and Cauchy distributions, and, sometimes, a mixture of the normal and Cauchy distributions, are suitable for modelling such financial data. The family of stable distributions can provide better modelling for such financial data sets. Financial data in high-volatility markets may be better modelled, in many cases, by the Linnik distribution in comparison to the stable distribution. This highly flexible family of distributions is better capable of modelling the inflection points and tail behaviour compared to the other existing models. The estimation of the tail thickness of heavy-tailed financial data is important in the context of modelling. However, the new probability distributions do not admit any closed analytical form of representation. Thus, novel methods need to be developed, as only a few can be found in the literature. Here, we recall a recent novel method, developed by the authors, based on a trigonometric moment estimator using circular distributions. The linear data may be transformed to yield circular data. This transformation is solely for yielding a suitable estimator. Our aim in this paper is to provide a review of the few existing methods, discuss some of their drawbacks, and also provide a universal ($\forall \alpha \in (0,2]$), efficient, and easily implementable estimator of $\alpha$ based on the transformation mentioned above. Novel, circular-statistics-based tests for the index parameter $\alpha$ of the stable and Linnik distributions are introduced and also exemplified with real-life financial data. Two real-life data sets are analysed to exemplify the methods recommended and enhanced by the authors.

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 $\alpha$ lies between 1 and 2. A similar approach can be adopted when $\alpha$ (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.

In the spirit of the solution of for modeling price changes in high-volatility markets for univariate commodities, here we generalize an approach to the case of modeling price changes jointly for two related commodities. Often, conditional distributions are more easily understood in financial markets, where the fluctuations in one commodity can shed significant light on the behavior of a related commodity. With this observation, we enhance and characterize the entire family of bivariate joint densities for which both the conditional distributions are specified to be of the Laplace form. Such bivariate distributions will be referred to as bivariate Laplace conditional (BLC) distributions. We study the marginals of the BLC distributions and establish that they are not only sub-Gaussian but also super-Laplacian and, hence, super-Cauchy, i.e., they have heavier tails than Gaussian distributions but lighter tails than the usual Laplace and Cauchy distributions. Distance correlation is suggested as a measure of the association between the two marginal variables, as their product moment correlation is zero but they may be non-linearly dependent. A real-life data set is analyzed to illustrate the use of BCL distributions in practice. We believe that this is the first work using conditional specifications in bivariate financial data analysis.