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

Part of: Stochastic processes Survival analysis and censored data Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

C. Costantini  and
F. Spizzichino
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.
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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.

Keywords

BURN-INOPTIMAL STOPPINGFREE BOUNDARY PROBLEMMONOTONE REGION

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62N05: Reliability and life testing
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 267 - 282
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'.

References

[1] Arjas, E. (1989) Survival models and martingale dynamics. Scand. J. Statist. 16, 177–225.Google Scholar
[2] Barlow, R. E. and Proschan, F. (1988) Life distribution models and incomplete data. In Handbook of Statistics. Vol. 7. ed Krishnaiah, P. R. and Rao, C. R. Elsevier, Amsterdam. pp. 225–249.Google Scholar
[3] Bergman, B. (1985) On reliability and its applications. Scand. J. Statist. 12, 1–41.Google Scholar
[4] Block, H. W. and Savits, T. (1994) Burn-in. Technical report. No 94-02. Department of Mathematics and Statistics, University of Pittsburgh.Google Scholar
[5] Brémaud, P. (1981) Point Processes and Queues. Springer, Berlin.CrossRefGoogle Scholar
[6] Clarotti, C. A. and Spizzichino, F. (1990) Bayes burn-in procedures. Prob. Eng. Inf. Sci. 4, 437–145.CrossRefGoogle Scholar
[7] Costantini, C. and Spizzichino, F. (1991) Optimal stopping of life testing; use of stochastic orderings in the case of conditionally exponential lifetimes. In Stochastic Orders and Decision under Risk. ed. Mosler, K. and Scarsini, M. IMS Monograph Series. pp. 95–103.CrossRefGoogle Scholar
[8] Del Grosso, G., Gerardi, A. and Koch, G. (1992) Optimality in accelerated life tests. Math. Operat. Res. 17, 866–881.CrossRefGoogle Scholar
[9] Jensen, F. and Petersen, N. E. (1981) Burn-in. Wiley, New York.Google Scholar
[10] Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.Google Scholar
[11] Nappo, G. and Spizzichino, F. (1990) The Markov process associated to exchangeable random variables. Preprint URLS-DM/NS-90/009. Department of Mathematics, University of Rome ‘La Sapienza’.Google Scholar
[12] Norros, I. (1986) A compensator representation of multivariate life length distributions with applications. Scand. J. Statist. 13, 99–112.Google Scholar
[13] Shepp, L. A. (1967) Explicit solutions of some problems of optimal stopping. Ann. Math. Statist. 38, 1912–1914.CrossRefGoogle Scholar
[14] Shiryaev, A. N. (1978) Optimal Stopping Rules. Springer, Berlin.Google Scholar
[15] Spizzichino, F. (1991) Sequential burn-in procedures. J. Statist. Plan. Inf. 29, 187–197.CrossRefGoogle Scholar