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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

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Durbin, J. (1971) Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.CrossRefGoogle Scholar
Epanechnikov, V. A. (1968) The significance level and power of the two-sided Kolmogorov test in the case of small sample sizes. Theory Prob. Appl. 13, 686–690.Google Scholar
Eplett, W. J. R. (1981) The inadmissibility of linear rank tests under Bahadur efficiency. Ann. Statist. 9, 1079–1086.Google Scholar
Lai, T. L. (1976) Boundary crossing probabilities for sample sums and confidence sequences. Ann. Prob. 4, 299–312.CrossRefGoogle Scholar
Robbins, H. and Siegmund, D. O. (1970) Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Statist. 41, 1410–1429.Google Scholar
Slepian, D. (1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.CrossRefGoogle Scholar