A Bivariate Semiparametric Control Chart Based on Order Statistics

In this article, a new bivariate semiparametric Shewhart-type control chart is presented. The proposed chart is based on the bivariate statistic (X(r), Y(s)), where X(r) and Y(s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well-known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in-control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart. An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ2 chart as well as to the nonparametric SN2 and SR2 charts that are based on the multivariate form of the sign test and the Wilcoxon signed-rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a real-world problem related to winery production is presented. Copyright © 2016 John Wiley & Sons, Ltd.

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