Bootstrap-assisted bisection

Manuscript in preparation

Zago D., Capizzi G., Qiu P. (202+). An Improved Bisection-type Algorithm for Control Chart Calibration. Submitted.

Description

The control limit calculation is a critical step when designing control charts in statistical process control. Traditional control chart designs require computing the control limits so that a characteristic of the in-control run length distribution, such as the mean or median, equals a pre-determined value. When the complexity of the IC process distribution makes analytical methods difficult, Monte Carlo approaches can be used to find the control limits.

To address the computational challenges of classical Monte-Carlo approaches, we introduce a modified bisection algorithm, referred to as the Bootstrap-Assisted bisection (BA-bisection) algorithm. The primary goal of the proposed algorithm is to overcome the shortcomings of the traditional bisection searching algorithm and extend its applicability to multi-chart scenarios.

Instead of approximating the in-control run length distribution as in the traditional bisection searching algorithm, the proposed algorithm uses bootstrap to estimate the IC distribution of the charting statistic at each time $t = 1, 2, \ldots, T$ during process monitoring. Consequently, a bisection search can be applied to the estimated IC distribution of the charting statistic with minimal computational cost to find the appropriate control limit values.

The new method eliminates the need to specify an initial range for searching. Additionally, an efficient generalization of this approach is proposed to handle the multi-chart setting. Numerical results confirm that our method offers an efficient and reliable way to compute control limits compared to the conventional bisection searching algorithm and the algorithm based on stochastic approximations.