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ARL-Unbiased CUSUM Schemes to Monitor Binomial Counts

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Frontiers in Statistical Quality Control 13 (ISQC 2019)

Part of the book series: Frontiers in Statistical Quality Control ((FSQC))

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Abstract

Counted output, such as the number of defective items per sample, is often assumed to have a marginal binomial distribution. The integer and asymmetrical nature of this distribution and the value of its target mean hinders the quality control practitioner from dealing with a chart for the process mean with a pre-stipulated in-control average run length (ARL) and the ability to swiftly detect not only increases but also decreases in the process mean. In this paper we propose ARL-unbiased cumulative sum (CUSUM) schemes to rapidly detect both increases and decreases in the mean of independent and identically distributed as well as first-order autoregressive (AR(1)) binomial counts. Any shift is detected more quickly than a false alarm is generated by these schemes and their in-control ARL coincide with the pre-specified in-control ARL. We use the R statistical software to provide compelling illustrations of all these CUSUM schemes.

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Notes

  1. 1.

    Notice that the transient states can be ordered as follows: \((0,0), (0,1/b^-), \dots , (0,c^-/b^-),\)

    \((1,0), (1,1/b^-), \dots , (1,c^-/b^-),\) \(\dots , (c^+/b^+,0), (c^+/b^+,1), \dots , (c^+/b^+,c^-/b^-)\).

  2. 2.

    The transient states can be ordered as follows: \((0, 0,0), (0, 0,1/b^-), \dots , (0, 0,c^-/b^-),\)\((0,1,0), (0,1,1/b^-), \dots , (0,1,c^-/b^-), \dots , (n,c^+/b^+,0), (n, c^+/b^+,1), \dots , (n, c^+/b^+,c^-/b^-)\).

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Acknowledgements

The first author acknowledges the financial support of the Portuguese FCT – Fundação para a Ciência e a Tecnologia, through the project UID/Multi/04621/2019 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, Universidade de Lisboa.

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Authors and Affiliations

  1. CEMAT (Center for Computational and Stochastic Mathematics) and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisbon, Portugal

    Manuel Cabral Morais

  2. Department of Mathematics and Statistics, Faculty of Economics and Social Sciences, Helmut Schmidt University, Hamburg, Postfach 700822, 22008, Hamburg, Germany

    Sven Knoth & Christian H. Weiß

  3. Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisbon, Portugal

    Camila Jeppesen Cruz

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  1. Manuel Cabral Morais
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  2. Sven Knoth
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  4. Christian H. Weiß
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Correspondence to Manuel Cabral Morais .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany

    Sven Knoth

  2. Department of Statistics, European University Viadrina, Frankfurt (Oder), Germany

    Wolfgang Schmid

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Morais, M.C., Knoth, S., Cruz, C.J., Weiß, C.H. (2021). ARL-Unbiased CUSUM Schemes to Monitor Binomial Counts. In: Knoth, S., Schmid, W. (eds) Frontiers in Statistical Quality Control 13. ISQC 2019. Frontiers in Statistical Quality Control. Springer, Cham. https://doi.org/10.1007/978-3-030-67856-2_6

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