A new EWMA chart for simultaneously monitoring the parameters of a shifted exponential distribution

In various scenarios where products and services are accompanied by warranties to ensure their reliability over a specified time, the two-parameter (shifted) exponential distribution serves as a fundamental model for time-to-event data. In modern production process, the products often come with warranties, and their quality can be manifested by the changes in the scale and origin parameters of a shifted exponential (SE) distribution. This paper introduces the Max-EWMA chart, employing maximum likelihood estimators and exponentially weighted moving average (EWMA) statistics, to jointly monitor SE distribution parameters. Additionally, we extend two additional charts, namely the Max-DEWMA and Max-TEWMA charts to enhance early-stage shift detection. Performance evaluations under zero-state and steady-state conditions compare these charts with the existing Max-CUSUM chart in terms of expected value and standard deviation of the run length (RL) distribution. Our findings reveal that among the Max-EWMA schemes, the Max-EWMA SE chart outperforms the others in terms of steady-state performance, while the Max-TEWMA chart surpasses the Max-EWMA and Max-DEWMA SE charts in respect to zero-state performance. Moreover, the proposed Max-EWMA schemes demonstrate advantages over Max-CUSUM, especially for small to moderate smoothing constants. We also provide an illustrative example to demonstrate the implementation of the proposed schemes.