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HNBUE survival under some shock models
Umeå University, Faculty of Science and Technology, Mathematical statistics.
1980 (English)Report (Other academic)
Abstract [en]

Suppose that a device is subjected to shocks governed by a counting processN = {N(t): t > 0}. The probability that the device survives beyond t is00then H(t) = E P(N(t) = k)P, , where P, is the probability of survivingk=0 _k shocks. In this paper we prove that H(t) is HNBUE (HNWUE), i.e.00 00/ H(x)dx < (>) y exp(-t/y) for t > 0, where y = / H(x)dx, under semet " 0 — 00different conditions on N and ^^^=0' ^or ^nstance we stuc^y the casewhen the interarrivai times between the shocks are independent and HNBUE(HNWUE). This situation includes the cases when N is a Poisson processor a stationary birth process. Further a certain cumulative damage modelis studied.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 1980. , 24 p.
Series
Statistical research report, ISSN 0348-0399 ; 1980:3
Keyword [en]
Shock model, Poisson process, birth process, survival function, HNBUE, HNWUE
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-78991OAI: oai:DiVA.org:umu-78991DiVA: diva2:638388
Projects
digitalisering@umu
Note

There are some occurring misspellings in the formulas in the abstract on this webpage. Read the abstract in the full-text document for correct spelling in formulas, see the downloadable file.

Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved
In thesis
1. Properties and tests for some classes of life distributions
Open this publication in new window or tab >>Properties and tests for some classes of life distributions
1980 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

A life distribution and its survival function F = 1 - F with finitemean y = /q F(x)dx are said to be HNBUE (HNWUE) if F(x)dx < (>)U exp(-t/y) for t > 0. The major part of this thesis deals with the classof HNBUE (HNWUE) life distributions. We give different characterizationsof the HNBUE (HNWUE) property and present bounds on the moments and on thesurvival function F when this is HNBUE (HNWUE). We examine whether theHNBUE (HNWUE) property is preserved under some reliability operations andstudy some test statistics for testing exponentiality against the HNBUE(HNWUE) property.The HNBUE (HNWUE) property is studied in connection with shock models.Suppose that a device is subjected to shocks governed by a counting processN = {N(t): t > 0}. The probability that the device survives beyond t isthen00H(t) = S P(N(t)=k)P, ,k=0where P^ is the probability of surviving k shocks. We prove that His HNBUE (HNWUE) under different conditions on N and * ^orinstance we study the situation when the interarrivai times between shocksare independent and HNBUE (HNWUE).We also study the Pure Birth Shock Model, introduced by A-Hameed andProschan (1975), and prove that H is IFRA and DMRL under conditions whichdiffer from those used by A-Hameed and Proschan.Further we discuss relationships between the total time on test transformHp^(t) = /q ^F(s)ds , where F \t) = inf { x: F(x) > t}, and differentclasses of life distributions based on notions of aging. Guided by propertiesof we suggest test statistics for testing exponentiality agains t IFR,IFRA, NBUE, DMRL and heavy-tailedness. Different properties of these statisticsare studied.Finally, we discuss some bivariate extensions of the univariate properties NBU, NBUE, DMRL and HNBUE and study some of these in connection with bivariate shock models.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 16 p.
Keyword
Life distribution, survival function, exponential distribution, IFR, IFRA, NBUE, DMRL, HNBUE, shock model, total time on test transform, testing of exponentiality
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78990 (URN)
Public defence
1980-10-17, Samhällsvetarhuset, hörsal D, Umeå universitet, Umeå, 09:15
Projects
digitalisering@umu
Note

There are some occurring misspellings in the formulas in the abstract on this webpage. Read the abstract in the full-text document for correct spelling in formulas, see the downloadable file.

Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved

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Citation style
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Output format
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