ARL‐Unbiased Exponential Control Charts With Estimated Parameters: Statistical Design and Implementation

ABSTRACT The bias of a control chart is an important factor in practical applications. Average run length (ARL)‐biased control charts have their out‐of‐control (OOC) ARL (ARL OOC ) greater than the in‐control (IC) ARL (ARL in ) for some values of the monitored parameter. This may cause a delayed detection of a true alarm (than a false alarm) and thus can have serious negative consequences in practice. To remedy this problem, ARL‐unbiased charts have been considered in literature but mostly in the known IC parameters case (case K). In this paper, Shewhart‐type conditional ARL‐unbiased (CARLU) charts are considered for the more practical problem of unknown IC parameters case (case U), with a focus on a deteriorating process. The popular exponential time between events (TBE) chart is considered for illustration of the ideas. Other charts can be considered along the same lines. The proposed charts are designed using both the unconditional and the conditional perspective, respectively. The required chart design constants are calculated. A published dataset is used to illustrate the methodology in a case study. Chart performance is examined via various metrics. Interestingly, it is found that tuning the chart design parameters can lead to substantial savings in the Phase I sample size requirement under the conditional perspective. To this end, some recommendations on minimum sample size are given. A summary and some concluding remarks are provided.