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Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

Published online by Cambridge University Press:  14 July 2016

W. J. R. Eplett
W. J. R. Eplett*
Affiliation:
University of Oxford
*
∗ Postal address: Mathematical Institute, 24–29 St. Giles', Oxford 0X1 3LB, U.K.
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Abstract

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

Keywords

IFRA/DFRA DISTRIBUTIONSSLEPIAN'S LEMMA
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 529 - 536
Copyright
Copyright © Applied Probability Trust 1983 

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References

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