Distribution-free EWMA schemes for simultaneous monitoring of time between events and event magnitude

Simultaneous monitoring of the time between events (TBEs) and event magnitude, such as, the time between successive manufacturing plant accidents and the magnitude of damage in an industry, for example, coal mines, has evolved as a popular research topic in industrial engineering. Most of the existing methods assume that the TBEs and magnitude are mutually independent, which is difficult to justify in practice. Several researchers consider some bivariate parametric models to take into account the dependence between the TBEs and magnitude, and construct joint monitoring schemes. In practice, such parametric methods are often unrealistic in absence of prior knowledge about the underlying distribution of a process. To this end, we use two distribution-free statistics, proposed respectively by Jurečková and Kalina (2012) and Mathur (2009), to design two exponentially weighted moving average schemes, abbreviated as EWMA-JKBWS and EWMA-Mathur. We offer detailed implementation strategies for these two schemes and provide simplified computational algorithm for determination of control limits. Moreover, we investigate both the IC and OOC performances in a comprehensive comparison study. The results indicate that in general, the EWMA-JKBWS scheme performs significantly better than the EWMA-Mathur scheme in large number of situations. Finally, we provide two real examples to illustrate the application of the proposed approach.