A General Framework for Generating Three-Components Heavy-Tailed Distributions with Application

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Abstract
The estimation of a certain threshold beyond which an extreme value distribution can be fitted to the tail of a data distribution remains one of the main issues in the theory of statistics of extremes. While standard Peak over Threshold (PoT) approaches determine this threshold graphically, we introduce in this paper a general framework which makes it possible for one to determine this threshold algorithmically by estimating it as a free parameter within a composite distribution. To see how this threshold point arises, we propose a general framework for generating three-component hybrid distributions which meets the need of data sets with right heavy-tail. The approach involves the combination of a distribution which can efficiently model the bulk of the data around the mean, with an heavy-tailed distribution meant to model the data observations in the tail while using another distribution as a link to connect the two. Some special examples of distributions resulting from the general framework are generated and studied. An estimation algorithm based on the maximum likelihood method is proposed for the estimation of the free parameters of the hybrid distributions. Application of the hybrid distributions to the S &P 500 index financial data set is also carried out.

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