# Properties of the weakest-link exponentiated weibull model

## Abstract

The exponentiated Weibull distribution can be used to model the tensile strength of carbon fibers and carbon fibrous composites. Under the assumption that a carbon fiber can be modeled as made up of independent links, the strength of the weakest link is the stregth of the entire fiber. If in addition we assume that the strength of each link can be modeled by an exponentiated Weibull distribution, the strength of an entire fiber has the chain-of-links or weakest-link exponentiated Weibull distribution. The purpose of this thesis is to study the parameters of the weakest-link exponentiated Weibull distribution, to find an appropriate method to calculate the maximum likelihood estimators (MLEs) for these parameters, and to find a relation between these parameters and the parameters of the exponentiated Weibull distribution so that the two distributions are stochastically equal. A minimization method is used to approximate the distance between the probability density functions (pdf's) of the weakest-link exponentiated Weibull and the exponentiated Weibull distributions. Programs in the statistical package R are used to calculate the distance between the pdf's and for the actual minimization of the approximation of the distance. A direct maximization method is used to find the maximum likelihood estimators for the parameters of the weakest-link exponentiated Weibull distribution. The function optim of the statistical package R is used to find the MLEs. An extended simulation study is conducted using a random sample of size n = 1000 from the weakest-link exponentiated Weibull distribution to find the MLEs of its parameters and to find the corresponding parameters of the exponentiated Weibull distribution so that the two distributions are stochastically close to each other. Several direct optimization methods (options to the R function optim) are examined to find the best approximation to the MLEs. Approximations to the MLEs are recursively used to get the next set of approximations to the MLEs so that a better approximation could be achieved. The two probability distributions are used as models to fit the Bader and Priest (1982) data that contain three different lengths of a fiber. Using two different methods of optimization, MLEs are found for the parameters of the weakest-link exponentiated Weibull and the parameters of the exponentiated Weibull distributions. The mean square errors for each length of a fiber are calculated using each of the two probability distributions, and then they are compared. Using the values of the parameters for the two distributions obtained in the extended simulation study mentioned above (under the assumption that the two distributions are stochastically close to each other) and using linear regression, the Pearson product moment correlation coefficient, and several two- and three-dimensional plots, possible linear relationships between the parameters of the two distributions are examined.