Guaranteed conditional ARL performance in the presence of autocorrelation

The Average Run Length (ARL) performance of a control chart conditional on parameter estimates from a Phase I study may deviate significantly from its designed performance. To circumvent this problem, the guaranteed conditional performance measure is proposed in the literature. Hence, one does not intend to reach a specified in-control ARL exactly, but to reach an ARL value greater than or equal to a specified in-control ARL with a certain probability. In the literature, the control chart design for a guaranteed conditional performance has only been discussed for the case of independent and identically distributed data. Therefore, a novel guaranteed conditional performance approach for an autocorrelated data generating process is proposed, assuming that the process parameters are unknown. The approach is exemplified by considering an AutoRegressive dependence structure of order 1, i.e., AR(1), and by designing individuals control chart. Appropriate bootstrap schemes are provided for implementation. These bootstrap schemes extend easily from order one to general order p for an AR(p) process. Through simulations, performance analyses of the proposed approach and a study about the robustness with respect to distributional assumptions are provided.