Thesis (MSc)--Stellenbosch University, 2004. / ENGLISH ABSTRACT: The Bayesian approach to decision-making is considered in this thesis for reliability/survival
models pertaining to a Weibull class of distributions. A generalised right censored sampling
scheme has been assumed and implemented. The Jeffreys' prior for the inverse mean lifetime
and the survival function of the exponential model were derived. The consequent posterior distributions
of these two parameters were obtained using this non-informative prior. In addition
to the Jeffreys' prior, the natural conjugate prior was considered as a prior for the parameter
of the exponential model and the consequent posterior distribution was derived. In many
reliability problems, overestimating a certain parameter of interest is more detrimental than underestimating
it and hence, the LINEX loss function was used to estimate the parameters and
their consequent risk measures. Moreover, the same analogous derivations have been carried
out relative to the commonly-used symmetrical squared error loss function. The risk function,
the posterior risk and the integrated risk of the estimators were obtained and are regarded in
this thesis as the risk measures. The performance of the estimators have been compared relative
to these risk measures. For the Jeffreys' prior under the squared error loss function, the
comparison resulted in crossing-over risk functions and hence, none of these estimators are
completely admissible. However, relative to the LINEX loss function, it was found that a correct
Bayesian estimator outperforms an incorrectly chosen alternative. On the other hand for
the conjugate prior, crossing-over of the risk functions of the estimators were evident as a result.
In comparing the performance of the Bayesian estimators, whenever closed-form expressions
of the risk measures do not exist, numerical techniques such as Monte Carlo procedures were
used. In similar fashion were the posterior risks and integrated risks used in the performance
compansons.
The Weibull pdf, with its scale and shape parameter, was also considered as a reliability model.
The Jeffreys' prior and the consequent posterior distribution of the scale parameter of the
Weibull model have also been derived when the shape parameter is known. In this case, the estimation process of the scale parameter is analogous to the exponential model. For the case
when both parameters of the Weibull model are unknown, the Jeffreys' and the reference priors
have been derived and the computational difficulty of the posterior analysis has been outlined.
The Jeffreys' prior for the survival function of the Weibull model has also been derived, when
the shape parameter is known. In all cases, two forms of the scalar estimation error have been
t:. used to compare as much risk measures as possible. The performance of the estimators were
compared for acceptability in a decision-making framework. This can be seen as a type of
procedure that addresses robustness of an estimator relative to a chosen loss function. / AFRIKAANSE OPSOMMING: Die Bayes-benadering tot besluitneming is in hierdie tesis beskou vir betroubaarheids- / oorlewingsmodelle
wat behoort tot 'n Weibull klas van verdelings. 'n Veralgemene regs gesensoreerde
steekproefnemingsplan is aanvaar en geïmplementeer. Die Jeffreyse prior vir die
inverse van die gemiddelde leeftyd en die oorlewingsfunksie is afgelei vir die eksponensiële
model. Die gevolglike aposteriori-verdeling van hierdie twee parameters is afgelei, indien hierdie
nie-inligtingge-wende apriori gebruik word. Addisioneel tot die Jeffreyse prior, is die
natuurlike toegevoegde prior beskou vir die parameter van die eksponensiële model en ooreenstemmende
aposteriori-verdeling is afgelei. In baie betroubaarheidsprobleme het die oorberaming
van 'n parameter meer ernstige nagevolge as die onderberaming daarvan en omgekeerd
en gevolglik is die LINEX verliesfunksie gebruik om die parameters te beraam tesame met
ooreenstemmende risiko maatstawwe. Soortgelyke afleidings is gedoen vir hierdie algemene
simmetriese kwadratiese verliesfunksie. Die risiko funksie, die aposteriori-risiko en die integreerde
risiko van die beramers is verkry en word in hierdie tesis beskou as die risiko maatstawwe.
Die gedrag van die beramers is vergelyk relatief tot hierdie risiko maatstawwe. Die
vergelyking vir die Jeffreyse prior onder kwadratiese verliesfunksie het op oorkruisbare risiko
funksies uitgevloei en gevolglik is geeneen van hierdie beramers volkome toelaatbaar nie. Relatief
tot die LINEX verliesfunksie is egter gevind dat die korrekte Bayes-beramer beter vaar
as die alternatiewe beramer. Aan die ander kant is gevind dat oorkruisbare risiko funksies van
die beramers verkry word vir die toegevoegde apriori-verdeling. Met hierdie gedragsvergelykings
van die beramers word numeriese tegnieke toegepas, soos die Monte Carlo prosedures,
indien die maatstawwe nie in geslote vorm gevind kan word nie. Op soortgelyke wyse is die
aposteriori-risiko en die integreerde risiko's gebruik in die gedragsvergelykings.
Die Weibull waarskynlikheidsverdeling, met skaal- en vormingsparameter, is ook beskou as 'n
betroubaarheidsmodel. Die Jeffreyse prior en die gevolglike aposteriori-verdeling van die skaalparameter
van die Weibull model is afgelei, indien die vormingsparameter bekend is. In hierdie geval is die beramingsproses van die skaalparameter analoog aan die afleidings van die eksponensiële
model. Indien beide parameters van die Weibull modelonbekend is, is die Jeffreyse
prior en die verwysingsprior afgelei en is daarop gewys wat die berekeningskomplikasies is
van 'n aposteriori-analise. Die Jeffreyse prior vir die oorlewingsfunksie van die Weibull model
is ook afgelei, indien die vormingsparameter bekend is. In al die gevalle is twee vorms van
die skalaar beramingsfoute gebruik in die vergelykings, sodat soveel as moontlik risiko maatstawwe
vergelyk kan word. Die gedrag van die beramers is vergelyk vir aanvaarbaarheid binne
die besluitnemingsraamwerk. Hierdie kan gesien word as 'n prosedure om die robuustheid van
'n beramer relatief tot 'n gekose verliesfunksie aan te spreek.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/50086 |
| Date | 12 1900 |
| Creators | Negash, Efrem Ocubamicael |
| Contributors | Mostert, Paul J., Stellenbosch University. Faculty of Economy and Management Sciences. Department of Statistics and Actuarial Science. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | xii, 105 pages : illustrations |
| Rights | Stellenbosch University |
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