Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 2, 229-255

THE EFFECTS OF AUTOCORRELATION ON THE $R$-CHART AND THE $S^{2}$-CHART

By

RAID W. AMIN, University of West Florida, Pensacola

WOLFGANG SCHMID, Europa-Universi\"{a}t Frankfurt (Oder)

And

OLIVER FRANK, Ulm Universit\"{a}t

SUMMARY. The average run length (ARL) properties of Shewhart Range ($R$) and $S^2$- charts are evaluated under assumptions of normality and autocorrelation. \ \ It is shown that these charts sometimes are very sensitive to departures from the independence case when control limits based on independent observations are used. The ARL of the $R$-chart is obtained through simulation, and the ARL of the $S^2$-chart is approximated numerically using results for quadratic forms and Laguerre polynomials. The correct control limits for the $R$-chart and $S^2$-chart are provided for the case of an autoregressive process of order 1, and regression models are provided for estimating the correct ARL based on control limits for the independence case. \ An interesting result shows that there exist certain sample sizes for which the in-control ARL of the $S^2$-chart is relatively unaffected under a moderately strong (positive) correlation structure. It is also shown that when the correct control limits are used, the $R$-chart is more efficient than the $S^2$-chart in the presence of strong autocorrelation. However, if the autocorrelation is weak, the $S^2$- chart turns out to be better.

AMS (1991) subject classification. 62N10.

Key words and phrases. Correlated data, average run length, statistical process control, time series.

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