A parametric bootstrap method of computation of control limits of charts for skewed distributions

This article is proposing a new parametric bootstrap method of evaluation of control limits of charts for asymmetrically distributed process measurements. The proposed method modifies the authors' approach to evaluation of control limits based on pseudorandom numbers generation (presented in [1]) by using the unbiased estimator of within-subgroup variation (pooled variance) at the step of evaluation of the distribution parameters hence decreasing the probability of false alarm in a process monitoring phase (Phase II). The use of an average statistic of within-subgroup variation increases the robustness of control limits (to the presence of exceptional variation) that allows applying the proposed method in a retrospective analysis (Phase I). The method does not require nonlinear transformations of data, which may complicate the technical interpretation and application of the results of analysis. The proposed parametric bootstrap method may be used to construct a control chart for any statistic when process measurements are distributed in accordance with any (one- or two-parameter) theoretical law. Quality variables with such distributions are found in many industries: telecommunications, electronics, mechanical engineering, etc. A comparison between the performance of charts for averages and standard deviations of the proposed method and the same charts of alternative methods1 in Phase II is made in terms of type-I and type-II error rates (using Monte Carlo simulations for process measurements distributed in accordance with the lognormal and the Weibull laws). The type-I error rates of charts for averages and standard deviations of the proposed method are closer to the required value than the type-I error rates of charts for averages and standard deviations of the alternative methods. The type-II error rates of these charts of the proposed method enable them to detect large shifts in the process location and variance very quickly.