An ARL-Unbiased Modified np-Chart for Autoregressive Binomial Counts

Morais Manuel Cabral (), Wittenberg Philipp () and Cruz Camila Jeppesen ()
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Morais Manuel Cabral: Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico — Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Wittenberg Philipp: Department of Mathematics and Statistics, Helmut Schmidt University, Holstenhofweg 85, 22043 Hamburg, Germany
Cruz Camila Jeppesen: Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Stochastics and Quality Control, 2023, vol. 38, issue 1, 11-24

Abstract: Independence between successive counts is not a sensible premise while dealing, for instance, with very high sampling rates. After assessing the impact of falsely assuming independent binomial counts in the performance of np-charts, such as the one with 3-σ control limits, we propose a modified np-chart for monitoring first-order autoregressive counts with binomial marginals. This simple chart has an in-control average run length (ARL) larger than any out-of-control ARL, i.e., it is ARL-unbiased. Moreover, the ARL-unbiased modified np-chart triggers a signal at sample t with probability one if the observed value of the control statistic is beyond the lower and upper control limits L and U. In addition to this, the chart emits a signal with probability γ L {\gamma_{L}} (resp. γ U {\gamma_{U}} ) if that observed value coincides with L (resp. U). This randomization allows us to set the control limits in such a way that the in-control ARL takes the desired value ARL 0 {\operatorname{ARL}_{0}} , in contrast to traditional charts with discrete control statistics. Several illustrations of the ARL-unbiased modified np-chart are provided, using the statistical software R and resorting to real and simulated data.

Keywords: Run Length; Integer-Valued Autoregressive Processes; Randomization Probabilities; Statistical Process Control (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1515/eqc-2022-0052

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