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Statistical analysis and simulation methods related to load-sharing models.
Umeå University, Faculty of Science and Technology, Mathematical statistics.
2000 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We consider the problem of estimating the reliability of bundles constructed of several fibres, given a particular kind of censored data. The bundles consist of several fibres which have their own independent identically dis-tributed failure stresses (i.e.the forces that destroy the fibres). The force applied to a bundle is distributed between the fibres in the bundle, accord-ing to a load-sharing model. A bundle with these properties is an example of a load-sharing system. Ropes constructed of twisted threads, compos-ite materials constructed of parallel carbon fibres, and suspension cables constructed of steel wires are all examples of load-sharing systems. In par-ticular, we consider bundles where load-sharing is described by either the Equal load-sharing model or the more general Local load-sharing model.

In order to estimate the cumulative distribution function of failure stresses of bundles, we need some observed data. This data is obtained either by testing bundles or by testing individual fibres. In this thesis, we develop several theoretical testing methods for both fibres and bundles, and related methods of statistical inference.

Non-parametric and parametric estimators of the cumulative distribu-tion functions of failure stresses of fibres and bundles are obtained from different kinds of observed data. It is proved that most of these estimators are consistent, and that some are strongly consistent estimators. We show that resampling, in this case random sampling with replacement from sta-tistically independent portions of data, can be used to assess the accuracy of these estimators. Several numerical examples illustrate the behavior of the obtained estimators. These examples suggest that the obtained estimators usually perform well when the number of observations is moderate.

Place, publisher, year, edition, pages
Umeå: Umeå University , 2000. , 29 p.
Keyword [en]
Non-parametric and parametric estimation, load-sharing models, asymptotic distribution, martingale, resampling, life testing, reliability
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-46772ISBN: 91-7191-965-1 (print)OAI: oai:DiVA.org:umu-46772DiVA: diva2:440682
Public defence
2001-02-02, MA 121, Umeå University, 14:03 (English)
Opponent
Supervisors
Available from: 2011-09-15 Created: 2011-09-13 Last updated: 2011-09-15Bibliographically approved
List of papers
1. Estimation of the reliability of systems described by the Daniels Load-Sharing Model
Open this publication in new window or tab >>Estimation of the reliability of systems described by the Daniels Load-Sharing Model
1999 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

We consider the problem of estimating the failure stresses of bundles (i.e. the tensile forces that destroy the bundles), constructed of several statisti-cally similar fibres, given a particular kind of censored data. Each bundle consists of several fibres which have their own independent identically dis-tributed failure stresses, and where the force applied on a bundle at any moment is distributed equally between the unbroken fibres in the bundle. A bundle with these properties is an example of an equal load-sharing sys-tem, often referred to as the Daniels failure model. The testing of several bundles generates a special kind of censored data, which is complexly struc-tured. Strongly consistent non-parametric estimators of the distribution laws of bundles are obtained by applying the theory of martingales, and by using the observed data. It is proved that random sampling, with replace-ment from the statistical data related to each tested bundle, can be used to obtain asymptotically correct estimators for the distribution functions of deviations of non-parametric estimators from true values. In the case when the failure stresses of the fibres are described by a Weibull distribution, we obtain strongly consistent parametric maximum likelihood estimators of the distribution functions of failure stresses of bundles, by using the complexly structured data. Numerical examples illustrate the behavior of the obtained estimators.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1999
Keyword
Non-parametric and parametric estimation, Equal Load-sharing models, asymptotic distribution, martingale, resam-pling, life testing, reliability
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-46724 (URN)
Available from: 2011-09-13 Created: 2011-09-12 Last updated: 2011-09-15Bibliographically approved
2. On Non-Parametric Estimation of Poission Point Processes Related to Failure Stresses of Fibres
Open this publication in new window or tab >>On Non-Parametric Estimation of Poission Point Processes Related to Failure Stresses of Fibres
2000 (English)Report (Other academic)
Abstract [en]

We consider statistical analysis of the reliability of fibres. The problem is to estimate the distribution law of random failure stresses of fibres (i.e. the critical level of stresses that destroy fibres) by using data obtained in a special kind of test, where several fibres are tested until they break. All new pieces resulting from this test will also be tested, if they are long enough. The test ends when all the remaining pieces are too short to be tested further. We refer to these as binary tree structured tests. We assume that the cumulative hazard function (c.h.f.) of the failure stresses of these fibres is continuous, and that the fibres are statistically identical. Under these assumptions we obtain, as the number of tested fibres increases, a strongly consistent Nelson-Aalen type estimator of the c.h.f. The functional central limit resampling theorem in Skorohod space is proved. It justifies the possibility of using resampling for estimating the accuracy of these estimators. The theorem shows that resampling can be used to asymptotically consistently estimate distribution laws of continuous functionals of the random deviations between the estimator and the true c.h.f.. For example, resampling can be used to estimate the distribution law of the maximum distance between estimators and estimands. Numerical examples suggest that resampling works well for a moderate number of tested fibres.

Series
Research report in mathematical statistics, ISSN 1653-0829 ; 2000-1
Keyword
load-sharing models, non-parametric esti-mation, binary tree structured tests, resampling, reliability, martingale, Skorohod space
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-46726 (URN)
Available from: 2011-09-12 Created: 2011-09-12 Last updated: 2011-09-15Bibliographically approved
3. Non-Parametric Estimators Related to Local Load-Sharing Models
Open this publication in new window or tab >>Non-Parametric Estimators Related to Local Load-Sharing Models
1999 (English)Report (Other academic)
Abstract [en]

We consider the problem of estimating the cumulative distribution function of failure stresses of bundles (i.e. the tensile forces that destroy bundles), constructed of several statistically similar fibres, given a particu-lar kind of censored data. Each bundle consists of several fibres which have their own independent identically distributed failure stresses, and where the force applied on a bundle at any moment is distributed between the fibres in the bundle according to the local load-sharing model.

The testing of several bundles generates a special kind of censored data, which is complexly structured. Consistent non-parametric estima-tors of the distribution laws of bundles are obtained by applying the theory of martingales, and by using the observed data. It is showed that ran-dom sampling, with replacement from the statistical data related to each tested bundle, can be used to estimate the accuracy of our non-parametric estimators. Numerical examples illustrate the behavior of the obtained es-timators.

Series
Research report in mathematical statistics, ISSN 1653-0829
Keyword
Non-parametric estimation, Load-sharing models, Local load-sharing models, martingale, resampling, life testing, reliability
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-46728 (URN)
Available from: 2011-09-12 Created: 2011-09-12 Last updated: 2011-09-15Bibliographically approved

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