Design of statistical process control methods for skewed populations with weighted standard deviations

Abstract

This thesis is concerned with the design of statistical process control methods - univariate/multivariate control charts and process capability indices (PCIs) - for skewed populations with weighted standard deviations (WSDs). This thesis is divided into the following four parts. (i) A univariate WSD method which approximates an arbitrary skewed univariate distribution with two normal distributions is proposed, and the method is applied to the univariate control charts - $\bar{X}$, cumulative sum, and exponentially weighted moving average charts. Control chart constants for skewed populations to estimate the standard deviation with subgroup ranges are derived, and an application example is given. The false alarm rates of these control charts are compared with those of standard and weighted variance (WV) control charts for lognormal, Weibull, and gamma distributions. (ii) Univariate PCIs based on the WSD method - $C^WSD_p$ and $C^WSD_pk$ - are proposed. The proposed WSD PCIs modify the standard PCIs by considering the standard deviations above and below the process mean separately. For symmetric populations, however, theses indices reduce to the standard PCIs. Methods of parameter estimation are provided, and a bias correction factor is suggested. Finite sample properties are investigated, and asymptotic distributions are also derived. The proposed WSD PCIs are compared with standard, Clements`, and WV PCIs in terms of nonconforming proportion. (iii) A multivariate WSD method which approximates a skewed multivariate distribution using segments of multivariate normal distributions is proposed, and the method is applied to the multivariate $T^2$ chart. This method reflects the skewness by adjusting the variance-covariance matrix with WSDs of each quality characteristic. Numerical comparisons between the WSD and standard $T^2$ charts are performed when the distribution is multivariate lognormal, Hougaard`s bivariate Weibull, or Cheriyan and Ramabhadran`s bivariate g...