This thesis is concerned with the design of statistical process control methods - $\bar{X}$ and $R$ control charts, variable sampling interval(VSI) $\bar{X}$ control charts, and process capability indices(PCIs)-using the weighted variance(WV) method with no assumptions on the population. This thesis is divided into the following three parts.
(ⅰ) A heuristic method based on the WV concept of setting up control limits of $\bar{X}$ and $R$ charts for skewed populations is proposed, and $\bar{X}$ and $R$ charts constants for skewed populations are obtained. This method provides asymmetric control limits in accordance with the direction and degree of skewness estimated from the sample data, by using different variances in computing upper and lower limits. For symmetric populations, however, these control limits are equivalent to those of Shewhart control charts. The new heuristic control charts are compared by Monte Carlo simulation with the Shewhart charts and the geometric control charts of Ferrell when the underlying distribution is Weibull or Burr's. The WV method is also compared with the exact method for the case where the underlying distribution is exponential.
(ⅱ) A modification of the WV $\bar{X}$ control charts in which the interval till the next sample varies depending on the current sample mean is considered. The proposed WV VSI $\bar{X}$ charts use long interval length if a sample mean falls between lower and upper threshold limits and short interval length if it falls outside the threshold limits but between the control limits. It provides asymmetric control limits from mean and asymmetric threshold limits from mode in accordance with the shape of the underlying population using different factors in computing upper and lower limits for skewed populations. When the underlying population is symmetric, however, the charts reduce to the standard VSI $\bar{X}$ charts. The performances of the WV VSI $\bar{X}$ charts are compared with the WV $\bar{X}$ charts and the standard VSI $\bar{X}$ charts when the underlying distribution is Weibull or lognormal.
(ⅲ) A method of constructing PCIs with no assumptions on the population is presented. This method adjusts the value of the indices according to the degree of skewness estimated from the sample data by considering the standard deviations above and below the process mean separately. For the symmetric populations, however, these indices are equivalent to the standard PCIs. An application example from a semiconductor manufacturing process is given. The asymptotic distributions of the estimators of the PCIs based on the WV method are obtained and performances of the estimators for moderate sample sizes are studied by Monte Carlo simulation.
