Finite‐Horizon EWMA Monitoring of the Ratio of Two Correlated Normal Variables in Short Production Runs

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ABSTRACT Monitoring the ratio of two correlated quality characteristics is difficult when only a small number of inspections is available before a scheduled changeover. This paper studies an upper‐sided exponentially weighted moving average chart, denoted EWMA‐, for the ratio of two correlated normal sample means over a finite production horizon. The monitored target is the parameter ratio , rather than the expectation of the sample ratio, because the unrestricted ratio of two normal variables need not possess a finite first moment. The cumulative distribution function of the ratio is evaluated through a bivariate‐normal probability representation and checked against the corrected closed‐form density. The continuous EWMA process is approximated by a finite‐state Markov chain whose first transition is evaluated exactly from the deterministic initial value . Across 400 designs recalculated with transient cells, the maximum absolute in‐control calibration residual was and all transition‐matrix row errors were at most . For the representative setting , , , and , the out‐of‐control TARL for a 10% upward shift decreased from 7.951 at to 2.405 at , while the within‐run signalling probability increased from 0.573 to greater than 0.999. Matched‐ comparisons with Shewhart and CUSUM ratio charts show that memory‐type charts substantially improve sensitivity to moderate persistent shifts, although no single memory‐type chart dominates across all reported designs. The study also reports complete run‐length probability mass functions, false‐alarm probabilities, convergence diagnostics, delayed‐change performance, and a machine‐readable reproducibility archive.

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