Bayesian analysis of process capability indices for single and multiple sources of variability

English: Process capability index (process performance index) -relates the specification limits to the performance of a process, it reduces complex information about the performance of a process to a single number. A capability index is a dimensionless measure of relative variability. In this thesis, Bayesian statistics is employed to simulate and estimate most of the widely used process capability indices. In Bayesian analysis, we assume that we have prior knowledge or information or opinion about parameters of a statistical distribution and very often in practice we do. We then attach a distribution to this belief. Parameters do not really have a distribution, parameters are constants, and so a prior distribution is a way of expressing our belief or opinion on our parameters. A posterior distribution is the belief distribution of the parameters after the outcomes of experiments (data) have been observed. There is now an updated belief distribution in light of the information from the data. Bayesian inference is shown to have a number of advantages. A full Bayesian analysis provides a natural way of taking into account all sources of uncertainty in the estimation of the parameters. Uncertainty about the true value of the process capability index is incorporated into the analysis through the choice of a prior distribution. The most familiar element of the Bayesian school is the use of the non-informative (objective) prior distribution, designed to be minimally informative in some sense. The most famous of these is the Jeffrey’s-rule prior and is utilised throughout the thesis. Scientists hold up objectivity as the ideal of science. Reference priors are a refinement of the Jeffrey’s-rule priors for higher dimensional problems that have proven to be remarkably successful. The probability matching prior is recommended because it is designed to produce posterior credible intervals which are asymptotically identical to their frequentist counterparts. The Bayesian simulation procedure employs the posterior distribution of the parameters in doing the simulations. The procedure is also shown to be useful and comparable to existing classical statistical procedures in solving the supplier selection problem. Data arising from multiple sources of variability are very common in practice. Virtually all industrial processes exhibit between-batch and within-batch components of variation. In some cases the between-batch (or between subgroup) component is viewed as part of the common-cause-system for the process. A process capability index in more general settings is developed using Cpl as a point of reference. Cpl is a single variance index and is adapted to give 2 and 3 variance components indices. The variance component model proves to be suitable for handling multiple sources of variability capability indices. Again, Bayesian simulation methods prove to be useful in handling these multiple sources of variability indices. Results show that the Bayesian simulation approach is just as good if not better than the standard classical statistics approach in assessing the capability of an industrial process. The added advantage of the Bayesian approach is that, from the posterior distribution of the capability indices, we are in a position to obtain quantiles, credible regions and perform other inferential tasks.