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Asymptotic behavior of a general class of mixture failure rates

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  01 July 2016

Maxim Finkelstein  and
Veronica Esaulova
Maxim Finkelstein*
Affiliation:
University of the Free State, Bloemfontein, and Max Planck Institute for Demographic Research
Veronica Esaulova*
Affiliation:
Otto-von-Guericke-Universität Magdeburg
*
∗ Postal address: Department of Mathematical Statistics, University of the Free State, PO Box 339, 9300 Bloemfontein, Republic of South Africa. Email address: finkelm.sci@mail.uovs.ac.za
∗∗ Postal address: Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany. Email address: veronica.esaulova@gmail.com
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Abstract

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Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Keywords

Mixture of distributionsdecreasing failure rateincreasing failure rateproportional hazards modelaccelerated life model

MSC classification

Primary: 62E10: Characterization and structure theory 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 244 - 262
Copyright
Copyright © Applied Probability Trust 2006 

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