Evaluation of Shewhart time-between-events-and-amplitude control charts for several distributions

In recent years, several control charts have been developed for the simultaneous monitoring of the time interval T and the amplitude X of events, denoted as the Time Between Events and Amplitude (TBEA) charts. A vast majority of these contributions focused on statistics based on the ratio (Formula presented.) and on a reduced number of potential distributions, that is, the exponential distribution for T and the normal distribution for X. In this article, we provide a comparison of the efficiency of the three different statistics, denoted as Z1, Z2, and Z3, suitable for monitoring TBEA data in the case of four distributions (gamma, lognormal, normal and Weibull) for both T and X. Some numerical evaluations are performed in order to compare the performance of these TBEA charts for detecting upward shifts under different scenarios. An illustrative example obtained from a French forest fire database is also provided to show the implementation of the proposed TBEA control charts.