Extreme Value Analysis for Mixture Models With Heavy-Tailed Impurity

This research is devoted to the extreme value analysis for mixture models with parameters depending on the number of available observations. Such models can be viewed as triangular arrays, for which there exist no general results concerning the asymptotic behaviour of maxima. The current study considers the case when one of the components has a heavy-tailed distribution and corresponds to the small mixing parameter, so that it can be associated with the "heavy-tailed impurity", which "pollutes" another component. We analyse two ways of modelling this "impurity", namely, by the non-truncated regularly varying law and its upper-truncated version with an increasing truncation level. The set of possible limit distributions for maxima turns out to be much more diverse than in the classical setting, especially for a mixture with the truncated component, where it includes four discontinuous laws not present in the Fisher-Tippett-Gnedenko theorem. For practical purposes we describe the procedure of application of the considered model to the analysis of price variations. Using the data on the returns of BMW shares in 2019, we demonstrate that the proposed model provides a reasonable fit for such quantities, becoming an efficient tool for the analysis of their maximal values.