Evaluation of Shewhart time-between-events-and-amplitude control charts for correlated data

In recent years, several techniques based on control charts have been developed for thesimultaneousmonitoring of the time intervalTand the amplitudeXof events, known as time-between-events-and-amplitude (TBEA) charts. However, the vast majority of the existing works have some limitations. First, they usually focus on statistics based on the ratioXT, and second, they only investigate a reduced number of potential distributions, that is, the exponential distribution forTand the normal distribution forX. Moreover, until now, very few research papers have considered the potential dependence betweenTandX. In this paper, we investigate three different statistics, denoted asZ(1),Z(2), andZ(3), for monitoring TBEA data in the case of three potential distributions (gamma, normal, and Weibull), for bothTandX, using copulas as a mechanism to model the dependence. An illustrative example considering times between machine breakdowns and associated maintenance illustrates the use of TBEA control charts.