Constrained stochastic design of a cautious-learning CUSUM chart with guaranteed in-control performance*

With limited Phase I data, parameter-estimation error can substantially degrade the performance of cumulative sum (CUSUM) control charts. Cautious parameter learning mitigates this difficulty by delaying parameter updates to reduce contamination of baseline estimates, but existing cautious-learning CUSUM charts still rely on fixed defaults or heuristic tuning rules. This paper applies a constrained stochastic design to a cautious-learning CUSUM chart for detecting mean shifts in a univariate normal process under a guaranteed in-control performance requirement. A nested stochastic-approximation procedure jointly optimizes the reference value and learning constants through an outer simultaneous perturbation stochastic approximation (SPSA) search and an inner stochastic approximation (SA) calibration of the finite-sample control-limit adjustment. Simulation results show that the proposed design generally yields shorter out-of-control run lengths than benchmark cautious-learning CUSUM charts while maintaining the required in-control protection. The proposed chart is illustrated on a semi-synthetic NIST ZARR13 sequence with an injected mean shift.