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Estimating X ¯ Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants

Author

Listed:
  • Paul Braden

    (Department of Industrial, Manufacturing, and Systems Engineering, Texas Tech University, Lubbock, TX 79409, USA)

  • Timothy Matis

    (Department of Industrial, Manufacturing, and Systems Engineering, Texas Tech University, Lubbock, TX 79409, USA)

  • James C. Benneyan

    (Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA)

  • Binchao Chen

    (Amazon.com Inc., Seatle, WA 98170, USA)

Abstract

Shewhart X ¯ control charts commonly used for monitoring the mean of a process may be inaccurate or perform poorly when the subgroup size is small or the distribution of the process variable is skewed. Truncated saddlepoint distributions can increase the accuracy of estimated control limits by including higher order moments/cumulants in their approximation, yet this distribution may not exist in the lower tail, and thus the lower control limit may not exist. We introduce a novel modification in which some usually truncated higher-order cumulants are re-expressed as functions of lower-order cumulants estimated from data in a manner that ensures the existence of the truncated saddlepoint distribution over the complete domain of the random variable. The accuracy of this approach is tested in cases where the cumulants are assumed either known or estimated from sample data, and demonstrated in a healthcare application.

Suggested Citation

  • Paul Braden & Timothy Matis & James C. Benneyan & Binchao Chen, 2022. "Estimating X ¯ Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants," Mathematics, MDPI, vol. 10(7), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:7:p:1044-:d:778638
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    References listed on IDEAS

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    1. Wang, Suojin, 1992. "General saddlepoint approximations in the bootstrap," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 61-66, January.
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