### Abstract

Recently, Kanagawa et al. have proposed a new control chart, based on the estimator of Kullback-Leibler information, which enables the user to monitor both changes of the mean and the variance simultaneously for normally distributed characteristics. This control chart, called the (x̄, s) control chart, is designed in conformity with the probability limit method, that is, the control limit is derived from specifying the probability of the first kind of error. However, they have not referred to the power by using this control chart to detect a departure of the process from the null population, i.e., in-control state. Therefore, we consider the power of the (x̄, s) control chart presented by Kanagawa et al. in this article. In the process, we identify the estimator of Kullback-Leibler information as the log-likelihood ratio statistic for the in-control state. Then, we first derive the cumulant generating function of the log-likelihood ratio statistic. Furthermore, we develop some approximations for the distribution of the log-likelihood ratio statistic, and we investigate the power of the (x̄, s) control chart.

Original language | English |
---|---|

Pages (from-to) | 928-951 |

Number of pages | 24 |

Journal | Naval Research Logistics |

Volume | 46 |

Issue number | 8 |

DOIs | |

Publication status | Published - Dec 1999 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Management Science and Operations Research

### Cite this

**Power of the (x̄, s) control chart based on the log-likelihood ratio statistic.** / Watakabe, Kyouko; Arizono, Ikuo.

Research output: Contribution to journal › Article

*Naval Research Logistics*, vol. 46, no. 8, pp. 928-951. https://doi.org/10.1002/(SICI)1520-6750(199912)46:8<928::AID-NAV4>3.0.CO;2-R

}

TY - JOUR

T1 - Power of the (x̄, s) control chart based on the log-likelihood ratio statistic

AU - Watakabe, Kyouko

AU - Arizono, Ikuo

PY - 1999/12

Y1 - 1999/12

N2 - Recently, Kanagawa et al. have proposed a new control chart, based on the estimator of Kullback-Leibler information, which enables the user to monitor both changes of the mean and the variance simultaneously for normally distributed characteristics. This control chart, called the (x̄, s) control chart, is designed in conformity with the probability limit method, that is, the control limit is derived from specifying the probability of the first kind of error. However, they have not referred to the power by using this control chart to detect a departure of the process from the null population, i.e., in-control state. Therefore, we consider the power of the (x̄, s) control chart presented by Kanagawa et al. in this article. In the process, we identify the estimator of Kullback-Leibler information as the log-likelihood ratio statistic for the in-control state. Then, we first derive the cumulant generating function of the log-likelihood ratio statistic. Furthermore, we develop some approximations for the distribution of the log-likelihood ratio statistic, and we investigate the power of the (x̄, s) control chart.

AB - Recently, Kanagawa et al. have proposed a new control chart, based on the estimator of Kullback-Leibler information, which enables the user to monitor both changes of the mean and the variance simultaneously for normally distributed characteristics. This control chart, called the (x̄, s) control chart, is designed in conformity with the probability limit method, that is, the control limit is derived from specifying the probability of the first kind of error. However, they have not referred to the power by using this control chart to detect a departure of the process from the null population, i.e., in-control state. Therefore, we consider the power of the (x̄, s) control chart presented by Kanagawa et al. in this article. In the process, we identify the estimator of Kullback-Leibler information as the log-likelihood ratio statistic for the in-control state. Then, we first derive the cumulant generating function of the log-likelihood ratio statistic. Furthermore, we develop some approximations for the distribution of the log-likelihood ratio statistic, and we investigate the power of the (x̄, s) control chart.

UR - http://www.scopus.com/inward/record.url?scp=0033354983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033354983&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1520-6750(199912)46:8<928::AID-NAV4>3.0.CO;2-R

DO - 10.1002/(SICI)1520-6750(199912)46:8<928::AID-NAV4>3.0.CO;2-R

M3 - Article

AN - SCOPUS:0033354983

VL - 46

SP - 928

EP - 951

JO - Naval Research Logistics

JF - Naval Research Logistics

SN - 0894-069X

IS - 8

ER -