Shewhart-Type TBEA Charts for Monitoring Frequency and Amplitude with Symmetry Structure Under Generalized Weibull and Generalized Log-Logistic Distributions

Control charts for monitoring time between events (T) and amplitude (X) have been developed in recent years. Many TBEA charts depend on limited models such as exponential, normal, and gamma distributions and mainly rely on the ratio statistic (XT). This representation ignores the symmetric relationship between event occurrence and event magnitude. This paper proposes Shewhart-type TBEA charts constructed from three statistics (Z1), (Z2), and (Z3) based on (X) and (T). The approach models symmetry between frequency and amplitude using generalized Weibull and generalized log-logistic distributions. The statistics maintain proportional invariance when both variables shift together, which enables balanced monitoring of the process. Several scenarios are examined for detecting upward shifts. Performance is assessed using numerical measures of detection efficiency and average run length. The results show improved detection compared with classical ratio-based TBEA charts. A real data example from a French forest fire database illustrates the ability of the proposed charts to detect simultaneous changes in occurrence rate and burn intensity.