Loss-based optimal control statistics for control charts

This work proposes a means for interconnecting optimal sample statistics with parameters of the process output distribution irrespective of the specific way in which these parameters change during transition to the out-of-control state (jumps, trends, cycles, etc). The approach, based on minimization of the loss incurred by the two types of decision errors, leads to a unique sample statistic and, therefore, to a single control chart. The optimal sample statistics are obtained as a solution of the developed optional boundary equation. The paper demonstrates that, for particular conditions, this equation leads to the same statistics as are obtained through the Neyman-Pearson fundamental lemma. Application examples of the approach when the process output distribution is Gamma and Weibull are given. A special loss function representing out-of-control state detection as a pattern recognition problem is presented.