ARL Estimation of the Control Chart of Log Likelihood Ratios’ Sum for Markov Sequence

Abstract

To evaluate the surveillance performance of a control chart with the charting statistic of the sum of log likelihood ratios in the statistical process control (SPC), in this paper, we give the proof procedure based on Markov chains for the asymptotic estimation of the average run length (ARL) for this kind of chart. The out-of-control ${\text{ARL}}_1$ is approximately equal to 1 for any fixed in-control ${\text{ARL}}_0$ with a negative control limit. By the equivalence between limit distribution of a sum and that of a suprema sum of Markov chain, we derive the estimation of ${\text{ARL}}_1$ with a large enough positive control limit. Numerical experiments are conducted to confirm our results.