Abstract Statistical surveillance is used to detect an important change in a process as soon as possible after it has occurred. The exponentially weighted moving averages (EWMA) method is used in industry, economics, and medicine. Three optimality criteria of surveillance are studied. The average run length (ARL) criterion violates commonly accepted inference principles, and the drawbacks are demonstrated. The expected delay criterion is based on the minimal expected delay from change to detection. The full likelihood ratio method is optimal according to this criterion. Approximations of this method turn out to be modifications of the EWMA method. The approximations lead to a formula for the optimal value of the smoothing parameter of the EWMA statistic. The usefulness of this formula is shown. It is demonstrated that, for EWMA, the minimax criterion agrees well with that of the ED criterion but not with that of the ARL criterion.