Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants
Shewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></in...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-03-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/7/1044 |
| _version_ | 1827623155106054144 |
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| author | Paul Braden Timothy Matis James C. Benneyan Binchao Chen |
| author_facet | Paul Braden Timothy Matis James C. Benneyan Binchao Chen |
| author_sort | Paul Braden |
| collection | DOAJ |
| description | Shewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> 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. |
| first_indexed | 2024-03-09T11:39:25Z |
| format | Article |
| id | doaj.art-99dffdb09f8b454b97fc0d9149a139bd |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:39:25Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-99dffdb09f8b454b97fc0d9149a139bd2023-11-30T23:36:21ZengMDPI AGMathematics2227-73902022-03-01107104410.3390/math10071044Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated CumulantsPaul Braden0Timothy Matis1James C. Benneyan2Binchao Chen3Department of Industrial, Manufacturing, and Systems Engineering, Texas Tech University, Lubbock, TX 79409, USADepartment of Industrial, Manufacturing, and Systems Engineering, Texas Tech University, Lubbock, TX 79409, USADepartment of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USAAmazon.com Inc., Seatle, WA 98170, USAShewhart <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/10/7/1044statistical process controlcontrol chartcumulant generating functionsaddlepoint approximationskewed probability distributions |
| spellingShingle | Paul Braden Timothy Matis James C. Benneyan Binchao Chen Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants Mathematics statistical process control control chart cumulant generating function saddlepoint approximation skewed probability distributions |
| title | Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants |
| title_full | Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants |
| title_fullStr | Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants |
| title_full_unstemmed | Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants |
| title_short | Estimating <inline-formula><math display="inline"><semantics><mover accent="true"><mi>X</mi><mo stretchy="false">¯</mo></mover></semantics></math></inline-formula> Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants |
| title_sort | estimating inline formula math display inline semantics mover accent true mi x mi mo stretchy false ¯ mo mover semantics math inline formula statistical control limits for any arbitrary probability distribution using re expressed truncated cumulants |
| topic | statistical process control control chart cumulant generating function saddlepoint approximation skewed probability distributions |
| url | https://www.mdpi.com/2227-7390/10/7/1044 |
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