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Frailty models based on Lévy processes

Part of: Probability theory on algebraic and topological structures Stochastic processes Survival analysis and censored data

Published online by Cambridge University Press:  22 February 2016

Håkon K. Gjessing ,
Odd O. Aalen  and
Nils Lid Hjort
Håkon K. Gjessing*
Affiliation:
University of Oslo and Norwegian Institute of Public Health
Odd O. Aalen*
Affiliation:
University of Oslo
Nils Lid Hjort*
Affiliation:
University of Oslo
*
∗ Postal address: Norwegian Institute of Public Health, PO Box 4404, Nydalen, N-0403 Oslo, Norway. Email address: hakon.gjessing@fhi.no
∗∗ Postal address: Section of Medical Statistics, University of Oslo, PO Box 1122, Blindern, N-0317 Oslo, Norway.
∗∗∗ Postal address: Department of Mathematics, University of Oslo, PO Box 1053, Blindern, N-0316 Oslo, Norway.
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Abstract

Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Lévy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function Lévy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.

Keywords

Frailtyquasi-stationarycompound Poisson processcumulative damage processLévy processhazard processpower variance function distribution

MSC classification

Primary: 62N01: Censored data models
Secondary: 60G51: Processes with independent increments; L'evy processes 60B10: Convergence of probability measures
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 532 - 550
Copyright
Copyright © Applied Probability Trust 2003 

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