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Explicit results on conditional distributions of generalized exponential mixtures

Part of: Special processes Distribution theory Distribution theory - Probability Operations research and management science

Published online by Cambridge University Press:  04 September 2020

Claudia Klüppelberg  and
Miriam Isabel Seifert
Claudia Klüppelberg*
Affiliation:
Technical University of Munich
Miriam Isabel Seifert*
Affiliation:
Ruhr University Bochum
*
*Postal address: Boltzmannstraße 3, 85748 Garching, Germany. Email: cklu@tum.de
**Postal address: Universitätsstraße 150, 44801 Bochum, Germany. Email: miriam.seifert@rub.de
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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.

Keywords

Conditional distributionconvolutionexponential distributiongeneralized exponential mixture distributionphase-type distributionsreliability theoryrisk managementtail behavior

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60E05: Distributions 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 760 - 774
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Copyright
© Applied Probability Trust 2020

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