Browsing by Author "Negash, Efrem Ocubamicael"

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    Risk and admissibility for a Weibull class of distributions
    (Stellenbosch : Stellenbosch University, 2004-12) Negash, Efrem Ocubamicael; Mostert, Paul J.; Stellenbosch University. Faculty of Economy and Management Sciences. Department of Statistics and Actuarial Science.
    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.