This talk has been automatically captioned. You can remove these by pressing CC on the video toolbar.

Name: Ekaterina Morozova

Talk Title: Extreme value analysis for mixture models with heavy-tailed impurity

Abstract: While there exists a well-established theory for the asymptotic behaviour of maxima of the i.i.d. sequences, very few results are available for the triangular arrays, when the distribution can change over time. Typically, the papers on this issue deal with convergence to the Gumbel law (see [1], [3]) or twice-differentiable distribution [4]. The current research is a joint work with Vladimir Panov, available as a preprint on arXiv.org [6]. It contributes to the aforementioned problem by providing the extreme value analysis for the mixture models with varying parameters, which can be viewed as triangular arrays. In particular, we consider the case of the heavy-tailed impurity, which appears when one of the components has a heavy-tailed distribution, and the corresponding mixing parameter tends to zero as the number of observations grows. 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. In the latter case, the resulting limit depends on the asymptotic behaviour of the truncation point, which is shown to be related to the truncation regimes introduced in [2]. For practical purposes we describe the procedure of the application of the considered model to the analysis of financial returns. In this context, our research develops the idea that their distribution is somewhere "in between" the exponential and power laws [5].

References:

[1] Anderson, C.W., Coles, S.G., and Hüsler, J. (1997). Maxima of Poisson-like variables and related triangular arrays. The Annals of Applied Probability, 7(4), 953–971.

[2] Chakrabarty, A. and Samorodnitsky, G. (2012). Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not? Stochastic models, 28(1), 109–143.

[3] Dkengne, P.S., Eckert, N., and Naveau, P. (2016). A limiting distribution for maxima of discrete stationary triangular arrays with an application to risk due to avalanches. Extremes, 19(1), 25–40.

[4] Freitas, A. V. and Hüsler, J. (2003). Condition for the convergence of maxima of random triangular arrays. Extremes, 6(4), 381-394.

[5] Malevergne, Y., Pisarenko, V. and Sornette, D. (2005). Empirical distributions of stock returns: between the stretched exponential and the power law? Quantitative Finance, 5(4), 379–401.

[6] Panov, V. and Morozova, E. (2021). Extreme value analysis for mixture models with heavy-tailed impurity. arXiv preprint arXiv:2103.07689.  

This talk is a contributed talk at EVA 2021.