An adaptive multivariate EWMA chart for monitoring Gumbel's bivariate exponential distributed data

In modeling the multivariate time between events (MTBE), Gumbel's Bivariate Exponential (GBE) distribution has played an important role in industrial or service processes. Some works have been conducted on monitoring the processes that follow the GBE distribution. However, existing works on monitoring the GBE distributed processes are mostly on constructing the chart for the specific change size, which actually may vary or not to be known in practice. This may cause the existing designed GBE monitoring schemes' poor detection performance for different changes. To overcome this limitation and improve the existing GBE charts' detection ability for different change sizes, this paper proposes a new multivariate exponentially weighted moving average (MEWMA) chart with an adaptive structure, named as AMEWMA, for monitoring the process following the GBE distribution. Monte Carlo simulation method is employed to obtain the run length (RL) properties, i.e., the average RL, standard deviation of RL, and median of RL, of the proposed monitoring scheme. By selecting different smoothing parameter, the charting parameters of the proposed AMEWMA GBE chart are obtained and the corresponding out-of-control RL performances are studied for different change sizes. A detailed comparative analysis is conducted between the proposed chart and some existing multivariate GBE charts. The findings indicate that the proposed AMEWMA GBE chart generally performs better than the competitors for all sizes of change, in terms of different RL measures. Moreover, in detecting a wide range of changes, it significantly outperforms its counterparts in terms of the RL's overall performance measures. Finally, a genuine dataset of patient headache relief times is utilized to demonstrate the application and execution of the AMEWMA GBE chart.