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

Manuel Cabral Morais; Philipp Wittenberg; Camila Jeppesen Cruz

doi:10.1515/eqc-2022-0052

Article

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

Manuel Cabral Morais , Philipp Wittenberg and Camila Jeppesen Cruz

Published/Copyright: March 31, 2023

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From the journal Stochastics and Quality Control Volume 38 Issue 1

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 (resp. γ 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 , 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

MSC 2020: 62P30

Funding source: Fundação para a Ciência e a Tecnologia

Award Identifier / Grant number: UIDB/04621/2020

Award Identifier / Grant number: UIDP/04621/2020

Funding statement: The first author acknowledges the financial support of the Portuguese FCT, Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico, Universidade de Lisboa.

Acknowledgements

We are most grateful to the reviewer who selflessly devoted his/her time to scrutinize this work and offered pertinent comments that led to an improved version of the original manuscript.

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Received: 2022-12-09

Revised: 2023-02-13

Accepted: 2023-02-20

Published Online: 2023-03-31

Published in Print: 2023-06-01

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Articles in the same Issue

https://doi.org/10.1515/eqc-2022-0052

Keywords for this article

Run Length; Integer-Valued Autoregressive Processes; Randomization Probabilities; Statistical Process Control