A Burr type X chain-of-links model
D'Ambrosio, Donna M.
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In this thesis, the study of the Burr type X (often abbreviated BurrX) family of distributions applied to the chain-of-links model is considered. The Burr type X distribution was introduced by I. W. Burr in 1942. At that time, Burr also introduced eleven other families of distributions to study failure type data. Of the twelve distributions he introduced, only two have garnered substantial attention. The Burr type X is one of those which has received attention. This attention is due to its flexibility and power in fitting many types of observed data. In this thesis, the chain-of-links Burr type X model is derived based on simple mathematical arguments. Once the form is found, various properties of the model are studied. Of particular interest is estimation of, and inference for, the parameters of the model. Estimation is accomplished using maximum likelihood estimation. A large-scale simulation is conducted to study the large sample behavior of the maximum likelihood estimates. The empirical distributions of the estimators are compared to the well-known normal distribution to determine if inference procedures based on the normal distribution are appropriate. If normal-based inference procedures are appropriate, the simulations will allow one to determine for what sample sizes they are applicable. Finally, the simulations will also allow one to determine what factors effect the degree to which the distribution of the maximum likelihood estimates deviate from the normal distribution. Bader and Priest (1982), of the University of Surrey in England, conducted several experiments on carbon fibrous composites. They studied the strengths of individual carbon fibers, and 1000 fiber collections held together by an epoxy resin known as a tow. The Weibull distribution is most often applied to this data set with mixed results. The Weibull model provides a good fit for the individual carbon fibers but provides a less-than-satisfactory fit for the tow data. It has been shown by Surles and Padgett (1998) that the Burr type X model provides good fits for both sets of data. In this thesis, the chain-of-links Burr type X model is applied to the Bader and Priest fiber data. Modified versions of the chain-of-links Burr type X model are also under study. These are known as the linear and power-law models. These models, if seen to be appropriate, allow for the study of a phenomenon known as the end-effect, or clamp effect. The act of testing a fiber generally weakens the ends of the fiber, here it is held in place on the testing device.