Statistical design of phase II exponential chart with estimated parameters under the unconditional and conditional perspectives using exact distribution of median run length

In this paper, we consider the q-quantiles of the conditional run length (CRL) distribution to design and evaluate the phase II exponential chart instead of the widely used metric, the average of CRL. The unconditional and conditional perspectives of the performance evaluation are considered to adjust the control limits of the phase II exponential chart based on the in-control (IC) q-quantiles of CRL (denoted by CQRL(q)) distribution. Under the unconditional approach, the mean of the IC CQRL(q), is set equal to some pre-specified value, say QRL(0) of the q-quantile of run length (QRL) whereas under the conditional perspective, the control limits are adjusted so that that the IC CQRL(q) meets or exceeds the nominal QRL(0) value with a high probability. The charts under both perspectives are designed so that they are QRL-unbiased for a nominal QRL(0) value. The IC and out-of-control (OOC) performance studies of the proposed charts are carried out based on the most promising quantile i.e. median run length (MRL). The study shows that the MRL-unbiased exponential chart under the conditional perspective has a better IC performance.