Explicit results on conditional distributions of generalized exponential mixtures

Abstract

For independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

Control charts for measurement error models

“…Golosnoy and Rossen 2018 ) or for the purpose of monitoring for changes in various types of networks (cf. Klüppelberg and Seifert 2019 , 2020 ; Chen et al 2022 ).…”

Characterization of valid auxiliary functions for representations of extreme value distributions and their max‐domains of attraction

“…VR and vMR apply for all MDA distributions with arbitrary , not only for as it is the case in regular variation approach. In particular, VR and vMR are often used for Gumbel MDA distributions, see Barbe & Seifert (2016), Dȩbicki et al (2018), Fougères & Soulier (2012), Hashorva (2012), Klüppelberg & Lindner (2005) and Klüppelberg & Seifert (2020) for VR, as well as Abdous et al (2008), Engelke et al (2019), Fasen et al (2006), Janßen (2010) and Seifert (2016) for vMR; this Gumbel case grants the most interesting analysis and results, as we will point out in the following sections. Using VR and vMR could be also convenient instead of applying the concept of regular variation suitable for Fréchet and Weibull MDA distributions characterized by parameter (cf.…”

Testing for parameter changes in linear state space models

“…Of course, our procedure is directly applicable for those multivariate SV models which could be written in a linear state‐space form, see, for example, Reference 39. Further, it could be also extended for analyzing financial networks with linear structure, as considered, for example, in References 40,41. In general, estimating stochastic volatility models based on realized volatility measures remains nontrivial even for univariate settings (see, e.g., References 33,42).…”