Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other Tools

This paper presents an improved version of the Shewhart-type generalized variance |S| control chart for multivariate Gaussian process dispersion monitoring, based on the Cornish-Fisher quantile formula for non-normality correction of the traditional normal based 3-sigma chart limits. Also, the exact sample distribution of |S| and its quantiles (chart exact limits) are obtained through the Meijer-G function (inverse Mellin-Barnes integral transform), and an auxiliary control chart based on the trace of the standardized S matrix is introduced in order to avoid non detection of certain changes in the process variance-covariance Σ matrix. The performance of the proposed CF-corrected control chart is compared, considering false alarm risk (using analytical and simulation tools), with the traditional normal based chart and with the exact distributed based chart (for dimensions d = 2 and d = 3). This study shows that the proposed control limit corrections do remove the drawback of excess of false alarm associated with the traditional normal based |S| control chart. The proposed new chart (with its corresponding auxiliary chart) is illustrated with two numerical examples.