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A modified exponentially weighted moving average (EWMA) scheme is a quality control chart which can quickly detect small shifts in autocorrelated data. In this article, an explicit formula is presented to calculate the average run length ( ARL ) for a modified EWMA chart with an autoregressive AR(p) model involving exponential white noise. Its performance is compared with a numerical integral equation method via a simulation study to determine whether the explicit formula can help decrease the processing time. Moreover, this scheme is tested against a classical EWMA control chart using the  ARL  and  RMI  results. Finally, the explicit formula for  ARL  is applied to a real numerical example from the health field to demonstrate the efficacy of our method.

Run length distribution for a modified EWMA scheme fitted with a stationary AR(p) model