TY - JOUR

T1 - Estimating ¯X Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants

AU - Braden, Paul

AU - Matis, Timothy

AU - Benneyan, James C.

AU - Chen, Binchao

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - 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.

AB - 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.

KW - control chart

KW - cumulant generating function

KW - saddlepoint approximation

KW - skewed probability distributions

KW - statistical process control

UR - http://www.scopus.com/inward/record.url?scp=85127562875&partnerID=8YFLogxK

U2 - 10.3390/math10071044

DO - 10.3390/math10071044

M3 - Article

AN - SCOPUS:85127562875

VL - 10

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 7

M1 - 1044

ER -