Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T09:12:00.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme-value properties of the explosion-time distribution in a pure birth process

Published online by Cambridge University Press:  14 July 2016

Paul D. Feigin  and
Emmanuel Yashchin
Paul D. Feigin*
Affiliation:
Technion — Israel Institute of Technology
Emmanuel Yashchin*
Affiliation:
Technion — Israel Institute of Technology
*
∗Postal address: Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Haifa 32000, Israel.
∗Postal address: Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Haifa 32000, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

In each of a large number N of independent cells a breakdown mechanism is under way and proceeds until the first of the cells actually fails. At such a time, in each cell, the situation reverts to some initial state and the mechanism restarts. In this paper we consider those mechanisms for which breakdown may be modelled as the explosion of a pure birth process. Of interest is the distribution of time between failures and the possibility of estimating N and/or model parameters by observing a sequence of failure times. Saddlepoint approximation methods are used in the relevant extreme-value theory analysis for two important cases.

Keywords

SADDLEPOINT APPROXIMATIONBREAKDOWN IN THIN FILM INSULATORSTAIL EQUIVALENCEEXTREME-VALUE THEORYORDER STATISTICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 500 - 509
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bromwich, T. J. I'A. (1965) Introduction to the Theory of Infinite Series. Macmillan, New York.Google Scholar
De Bruijn, N. G. (1953) The difference-differential equation F'(X) = eax+ßF (x − 1). Indag. Math. 15, 449–458.Google Scholar
De Bruijn, N. G. (1970) Asymptotic Methods in Analysis. North-Holland, Amsterdam.Google Scholar
De Haan, L. (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.Google Scholar
Dingle, R. B. (1973) Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London.Google Scholar
Doetsch, G. (1974) Introduction to the Theory and Application of the Laplace Transformation. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Hirschman, I. I. and Widder, D. V. (1955) The Convolution Transform. Princeton University Press, Princeton.Google Scholar
Kadary, V. and Klein, N. (1980) Electrical breakdown during the anodic growth of tantalum pentoxide. J. Electrochemical Soc. 127, 139–151.Google Scholar
Murray, J. D. (1974) Asymptotic Analysis. Clarendon Press, Oxford.Google Scholar
Resnick, S. I. (1971) Tail equivalence and its applications. J. Appl. Prob. 8, 136–156.Google Scholar
Weissman, I. (1978) Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73, 812–815.Google Scholar
Yashchin, E. (1981) Modelling and Analysing Breakdown Phenomena in Thin Film Insulators — A Stochastic Approach. Doctoral Dissertation, Technion, Haifa, Israel.Google Scholar