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Explicit solution of an optimal stopping problem: the burn-in of conditionally exponential components

Published online by Cambridge University Press:  14 July 2016

C. Costantini*
Affiliation:
Università di Chieti, ‘G. D'Annunzio’
F. Spizzichino*
Affiliation:
Università di Roma ‘La Sapienza‘
*
Postal address: Dipartimento di Sciente, Università di Chieti, ‘G. D'Annunzio', Viale Pindaro, 42–65127 Pescara, Italy.
∗∗Postal address: Dipartimento di Matematica, Università di Roma ‘La Sapienza' P.le A. Mero, 2–00185 Roma, Italy.

Abstract

We consider the problem of the optimal duration of a burn-in experiment for n identical units with conditionally exponential life-times of unknown parameter Λ. The problem is formulated as an optimal stopping problem for a suitably defined two-dimensional continuous-time Markov process. By exploiting monotonicity properties of the statistical model and of the costs we prove that the optimal stopping region is monotone (according to an appropriate definition) and derive a set of equations that uniquely determine it and that can be easily solved recursively. The optimal stopping region varies monotonically with the costs. For the class of problems corresponding to a prior distribution on Λ of type gamma it is shown how the optimal stopping region varies with respect to the prior distribution and with respect to n.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

This research was supported by CNR project ‘Statistica Bayesiana e simulazione in affidabilita' e modellistica biologica' and by MURST project ‘Processi aleatori e modelli stocastici-teoria e applicazioni alle scienze ad all'industria'.

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