Some robust approaches based on copula for monitoring bivariate processes and component-wise assessment

In this paper, we develop two adaptive approaches for detecting the signal source in a bivariate process when a shift occurs in the location vector or the scale matrix or both. The proposed method capitalises the notion of Sklar's principle of expressing any multivariate joint distribution in terms of univariate marginal-distribution functions and a copula, which represents the dependence structure between the variables. Motivated by this, we recommend monitoring the two marginal distributions and the copula function simultaneously using appropriate nonparametric (distribution-free) test statistics. At each stage of Phase-II monitoring, we adopt the permutation method for computing the individual p-values and derive the plotting statistics of our proposed schemes combining suitable transforms of the three p-values of the component testing. We establish the in-control robustness of the proposed surveillance plans and compare them with two competitors in terms of run length properties. Performance of the proposed schemes in detecting a correct out-of-control signal is as good or better than some existing charting schemes for bivariate process monitoring. The novelty of our proposed technique lies in the fact that it indigenously helps in identifying the component(s) responsible for the signal, which is not straightforward with the traditional schemes for surveillance of a bivariate process. Numerical results substantiate that the proposed procedure performs significantly better than its competitors in many cases. Also, we investigate the percentage of correct diagnosis of a signal via the proposed charting schemes. Nowadays, in monitoring and control of smooth service operations, the use of quality monitoring has increased than ever before, but the problem and data structures become more complicated in the Industry 4.0 era. We analyse two real case studies, one in the context of monitoring the response time and service quality in a call centre and the other related to the inspection of product quality, to illustrate the application of the proposed schemes.