Adaptive Cumulative SUM charts (ACUSUM) have been recently proposed for providing an overall good detection over a range of mean shift sizes. The basic idea of the ACUSUM chart is to first adaptively update the reference value based on an Exponentially Weighted Moving Average (EWMA) estimate and then to assign a weight on it using a certain type of weighting function. A linear weighting function is proposed that is motivated by likelihood ratio testing concepts and that achieves superior detection performance. Moreover, in view of the lower efficiency in tracking relative large mean shifts of the EWMA estimate, a generalized EWMA estimate is proposed as an alternative. A comparison of run length performance of the proposed ACUSUM scheme and other control charts is shown to be favorable to the former.