Risk and admissibility for a Weibull class of distributions

[摘要] ENGLISH ABSTRACT: The Bayesian approach to decision-making is considered in this thesis for reliability/survivalmodels pertaining to a Weibull class of distributions. A generalised right censored samplingscheme has been assumed and implemented. The Jeffreys' prior for the inverse mean lifetimeand the survival function of the exponential model were derived. The consequent posterior distributionsof these two parameters were obtained using this non-informative prior. In additionto the Jeffreys' prior, the natural conjugate prior was considered as a prior for the parameterof the exponential model and the consequent posterior distribution was derived. In manyreliability problems, overestimating a certain parameter of interest is more detrimental than underestimatingit and hence, the LINEX loss function was used to estimate the parameters andtheir consequent risk measures. Moreover, the same analogous derivations have been carriedout 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 inthis thesis as the risk measures. The performance of the estimators have been compared relativeto these risk measures. For the Jeffreys' prior under the squared error loss function, thecomparison resulted in crossing-over risk functions and hence, none of these estimators arecompletely admissible. However, relative to the LINEX loss function, it was found that a correctBayesian estimator outperforms an incorrectly chosen alternative. On the other hand forthe 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 expressionsof the risk measures do not exist, numerical techniques such as Monte Carlo procedures wereused. In similar fashion were the posterior risks and integrated risks used in the performancecompansons.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 theWeibull 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 casewhen both parameters of the Weibull model are unknown, the Jeffreys' and the reference priorshave 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, whenthe shape parameter is known. In all cases, two forms of the scalar estimation error have beent:. used to compare as much risk measures as possible. The performance of the estimators werecompared for acceptability in a decision-making framework. This can be seen as a type ofprocedure that addresses robustness of an estimator relative to a chosen loss function.

[发布机构] Stellenbosch University