Optimal Cusum Control Chart for Censored Reliability Data with Log-logistic Distribution
Sadeghpour Gildeh Bahram, Taghizadeh Maryam
Department of Statistics, Faculty of Mathematical Science
Ferdowsi University of Mashhad, Mashhad, Iran
*E-mail: sadeghpour@um.ac.ir m.taghizadeh@stu.umz.ac.ir
Received:
Received: 30 October 2014; revised: 09 October 2015; accepted: 20 October 2015; published online: 16 December 2015
DOI: 10.12921/cmst.2015.21.04.006
Abstract:
The goal of this work is to detect any potentially harmful change in a process. The reliability tests are assumed to
generate type-I right-censored data following a log-logistic distribution with scale parameter (η) and shape parameter (β). For this purpose, we have constructed a likelihood ratio based simultaneous cumulative sum (CUSUM) control chart that targets changes in both the failure mechanism and the characteristic life (the simultaneous CUSUM chart for detecting shifts in the shape and the scale parameters). This control chart displays best performance for combinations with larger positive or negative shifts in the shape parameter, signaling on average in 5 samples in an out-of-control situation, while targeting an in-control average run length of 370. The simultaneous CUSUM chart’s performance is highly dependent on the values of β, and on the interaction between them and the censoring rates and shift sizes.
Key words:
Average run length, Cumulative sum (CUSUM) chart, Likelihood ratio, Log-logistic distribution, Type-I right-censoring
References:
[1] W. Q. Meeker and L. Escobar, Statistical Methods for Relia-
bility Data, John Wiley and Sons, New York, NY 1998.
[2] D. M. Hawkins and D. H. Olwell, Cumulative Sum Charts
and Charting for Quality Improvement, Springer-Verlag, New
York, NY 1998.
[3] J. O’Quigley and L. Struthers, Survival models based upon
the logistic and log-logistic distributions, Computer Programs
in Biomedical Researches 15, 3-12 (1982).
[4] A. Ragab and J. Green, On order statistics from the log-logistic
distribution and their properties, Commun. Statist, Theory
and Meth. 13. 2713- 2724 (1984).
[5] S. Bennett, Progressive Censoring: Theory, Methods and Log-
logistic regression model for survival data, Applied Statistics
32, 165-171 (1983).
[6] E. Page, Continuous Inspection Schemes, Biometrika 41. 100-
115 (1954).
[7] D.C. Montgomery, Introduction to Statistical Quality Control,
5th edition, John Wiley and Sons, New York, NY 2004.
[8] G. Moustakides, Optimal Stopping Times for Detecting
Changes in Distributions, The Annals of Statistics 14(4), 1379-
1387 (1986).
[9] A. F. Bissell, CUSUM techniques for quality control (with
discussion), Applied Statistics 18, 1-30 (1969).
[10] D. A. Olteanu, Cumulative Sum Control Charts for Censored
Reliability Data, PhD thesis Faculty of the Virginia Polytech-
nic Institute and State University, Blacksburg, Virginia (2010).
The goal of this work is to detect any potentially harmful change in a process. The reliability tests are assumed to
generate type-I right-censored data following a log-logistic distribution with scale parameter (η) and shape parameter (β). For this purpose, we have constructed a likelihood ratio based simultaneous cumulative sum (CUSUM) control chart that targets changes in both the failure mechanism and the characteristic life (the simultaneous CUSUM chart for detecting shifts in the shape and the scale parameters). This control chart displays best performance for combinations with larger positive or negative shifts in the shape parameter, signaling on average in 5 samples in an out-of-control situation, while targeting an in-control average run length of 370. The simultaneous CUSUM chart’s performance is highly dependent on the values of β, and on the interaction between them and the censoring rates and shift sizes.
Key words:
Average run length, Cumulative sum (CUSUM) chart, Likelihood ratio, Log-logistic distribution, Type-I right-censoring
References:
[1] W. Q. Meeker and L. Escobar, Statistical Methods for Relia-
bility Data, John Wiley and Sons, New York, NY 1998.
[2] D. M. Hawkins and D. H. Olwell, Cumulative Sum Charts
and Charting for Quality Improvement, Springer-Verlag, New
York, NY 1998.
[3] J. O’Quigley and L. Struthers, Survival models based upon
the logistic and log-logistic distributions, Computer Programs
in Biomedical Researches 15, 3-12 (1982).
[4] A. Ragab and J. Green, On order statistics from the log-logistic
distribution and their properties, Commun. Statist, Theory
and Meth. 13. 2713- 2724 (1984).
[5] S. Bennett, Progressive Censoring: Theory, Methods and Log-
logistic regression model for survival data, Applied Statistics
32, 165-171 (1983).
[6] E. Page, Continuous Inspection Schemes, Biometrika 41. 100-
115 (1954).
[7] D.C. Montgomery, Introduction to Statistical Quality Control,
5th edition, John Wiley and Sons, New York, NY 2004.
[8] G. Moustakides, Optimal Stopping Times for Detecting
Changes in Distributions, The Annals of Statistics 14(4), 1379-
1387 (1986).
[9] A. F. Bissell, CUSUM techniques for quality control (with
discussion), Applied Statistics 18, 1-30 (1969).
[10] D. A. Olteanu, Cumulative Sum Control Charts for Censored
Reliability Data, PhD thesis Faculty of the Virginia Polytech-
nic Institute and State University, Blacksburg, Virginia (2010).
