Simultaneously monitoring frequency and magnitude of events based on bivariate gamma distribution

The occurrence of an event, for example, some mishaps in manufacturing processes or natural disasters such as floods or earthquakes, is often characterized by its frequency and magnitude. Procedures for simultaneously monitoring the event frequency and the event magnitude usually assume that the frequency and magnitude are two independent variables. However, the dependence between frequency and magnitude is very common in practice. In this paper, a bivariate gamma distribution is considered for modelling the event frequency and the magnitude with certain dependence structure. Based on this bivariate gamma distribution, a multivariate exponentially weighted moving average (MEWMA) procedure is designed for jointly monitoring the shifts in mean values of the frequency and the magnitude of an event. Some comparisons are carried out via Monte-Carlo simulations. The results show that our proposed MEWMA procedure has significant performance advantages in many situations for different shift domains. A realistic example is given to illustrate the construction mechanism.