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Properties and tests for some classes of life distributions
Umeå University, Faculty of Science and Technology, Mathematical statistics.
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 [en]
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: urn:nbn:se:umu:diva-78990OAI: oai:DiVA.org:umu-78990DiVA: diva2:638387
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
List of papers
1. Some properties of the HNBUE and HNWUE classes of life distributions
Open this publication in new window or tab >>Some properties of the HNBUE and HNWUE classes of life distributions
1980 (English)Report (Other academic)
Abstract [en]

The HNBUE (HNWUE) class of life distributions (i.e. for which f F (x)dx< (>)00 t< (>) y exp(-t/y) for t > 0, where y = / F(x)dx) is studied. We prove0that the HNBUE (HNWUE) class is larger than the NBUE (NWUE) class. We alsopresent some characterizations of the HNBUE (HNWUE) property by using theTotal Time on Test (TTT-) transform and the Laplace transform. Further weexamine whether the HNBUE (HNWUE) property is preserved under the reliabilityoperations (1) formation of coherent structure, (2) convolution and(3) mixture. Some bounds on the moments and on the survival function of aHNBUE (HNWUE) life distribution are also presented. The class of distributionswith the discrete HNBUE (discrete HNWUE) property (i.e. for which00 00 00I P. < (>) y(l-l/y)k for k = 0,1,2j=k J "where yi=0 JI p. and P. = E p, )J k=j+l kis also studied.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 48 p.
Series
Statistical research report, ISSN 0348-0399 ; 1980:8
Keyword
Life distribution, survival function, survival probability, HNBUE, HNWUE, discrete HNBUE, discrete HNWUE, TTT-transform, Laplace transform, damage model, coherent structure, convolution, mixture
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78994 (URN)
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
2. HNBUE survival under some shock models
Open this publication in new window or tab >>HNBUE survival under some shock models
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
Shock model, Poisson process, birth process, survival function, HNBUE, HNWUE
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78991 (URN)
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
3. IFRA or DMRL survival under the pure birth shock process
Open this publication in new window or tab >>IFRA or DMRL survival under the pure birth shock process
1980 (English)Report (Other academic)
Abstract [en]

Suppose that a device is subjected to shocks and that P^, k • 0, 1, 2,00denotes the probability of surviving k shocks. Then H(t) = E P(N(t) = k)P,k=0is the probability that the device will survive beyond t, where N = (N(t): t > 0} is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the Pure Birth Shock Model. In this paper we shall prove that H(t) is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 9 p.
Series
Statistical research report, ISSN 0348-0399 ; 1980:4
Keyword
Shock model, birth process, survival function, variation diminishing property, total positivity, IFRA, DFRA, DMRL, IMRL
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78992 (URN)
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
4. Some tests against aging based on the total time on test transform
Open this publication in new window or tab >>Some tests against aging based on the total time on test transform
1980 (English)Report (Other academic)
Abstract [en]

Let F be a life distribution with survival function F = 1 - F and00 —finite mean y * j n F(x)dx. The scaled total time on test transform-1F( t ) —<P,,(t) = /n /F(x)dx/y was introduced by Barlow and Campo (1975) as ar Utool in the statistical analysis of life data. The properties IFR, IFRA, NBUE, DMRL and heavy-tailedness can be translated to properties of tp„(t).rWe discuss the previously known of these relationships and present some new results. Guided by properties of <P„(t) we suggest some test statisticsrfor testing exponentiality against IFR, IFRA, NBUE, DMRL and heavy-tailed-ness, respectively. The asymptotic distributions of the statistics are derived and the asymptotic efficiencies of the tests are studied. The power for some of the tests is estimated by simulation for some alternatives when the sample size is n = 10 or n = 20.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 40 p.
Series
Statistical research report, ISSN 0348-0399 ; 1979:4
Keyword
Life distribution, survival function, aging, exponential distribution, IFR, IFRA, NBUE, DMRL, hypothesis testing, efficiency, consistency, power
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78993 (URN)
Projects
digitalisering@umu
Note

This is a revised version of Statistical Research Report 1979-4, Department of Mathematical Statistics, University of Umeå.

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
5. Testing exponentiality against HNBUE
Open this publication in new window or tab >>Testing exponentiality against HNBUE
1980 (English)Report (Other academic)
Abstract [en]

Let F be a life distribution with survival function F = 1 - F and00 —finite mean y = /q F(x)dx. Then F is said to be harmonic new better00 —than used in expectation (HNBUE) if / F(x)dx < y exp(-t/y) for t > 0. If the reversed inequality is true F is said to be HNWUE (W = worse). We develop some tests for testing exponentiality against the HNBUE (HNWUE) property. Among these is the test based on the cumulative total time on test statistic which is ordinarily used for testing against the IFR (DFR) alternative. The asymptotic distributions of the statistics are discussed. Consistency and asymptotic relative efficiency are studied. A small sample study is also presented.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 26 p.
Keyword
Life distribution, HNBUE, HNWUE, exponential distribution, TTT-transform, hypothesis testing, efficiency, consistency, power
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78995 (URN)
Projects
digitalisering@umu
Note

This is a revised version of Sections 5 and 6 in Statistical Research Report 1979-9, Department of Mathematical Statistics, University of Umeå.

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
6. On some classes of bivariate life distributions
Open this publication in new window or tab >>On some classes of bivariate life distributions
1980 (English)Report (Other academic)
Abstract [en]

During the last years efforts have been made in order to define suitable bivariate and multivariate extensions of the univariate IFR, IFRA, NBU NBUE and DMRL classes (with duals) of life distributions. In this paper we suggest two new bivariate NBUE (NWUE) and several bivariate HNBUE (HNWUE) definitions. Furthermore, we discuss some of the classes of multivariate life distributions proposed by Buchanan and Singpurwalla (1977). We also study two bivariate shock models. Suppose that two devices are subjected to shocks of some kind. Let P(k^,k2), k^,k2 = 0,1,2,..., denote the probability that the devices survive k^ and k2 shocks, respectively, and let T. denote the time to failure of device number j, j = 1,2, and let H(t^,t2) = P(T^ > t^,T2 > t2)• We study the shock models by Marshall and Olkin and by Buchanan and Singpurwalla and give sufficient conditions, containing P(k^,k2), k^,k2 = 0,1,2,..., under which H.(t^,t2) is bivariate NBU (NWU), bivariate NBUE (NWUE) and bivariate HNBUE (HNWUE) of different forms.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 50 p.
Series
Statistical research report, ISSN 0348-0399 ; 1980:9
Keyword
Life distribution, survival function, bivariate exponential distribution, bivariate geometric distribution, bivariate NBU, bivariate NBUE, bivariate HNBUE, bivariate shock models
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78996 (URN)
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
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf