On Shewhart Control Charts for Zero-Truncated Negative Binomial Distributions

Anwer Khurshid, Ashit B Chakraborty

Abstract


The negative binomial distribution (NBD) is extensively used for the
description of data too heterogeneous to be fitted by Poisson
distribution. Observed samples, however may be truncated, in the
sense that the number of individuals falling into zero class cannot be
determined, or the observational apparatus becomes active when at
least one event occurs. Chakraborty and Kakoty (1987) and
Chakraborty and Singh (1990) have constructed CUSUM and
Shewhart charts for zero-truncated Poisson distribution respectively.
Recently, Chakraborty and Khurshid (2011 a, b) have constructed
CUSUM charts for zero-truncated binomial distribution and doubly
truncated binomial distribution respectively. Apparently, very little
work has specifically addressed control charts for the NBD (see, for
example, Kaminsky et al., 1992; Ma and Zhang, 1995; Hoffman, 2003;
Schwertman. 2005).

The purpose of this paper is to construct Shewhart control charts
for zero-truncated negative binomial distribution (ZTNBD). Formulae
for the Average run length (ARL) of the charts are derived and studied
for different values of the parameters of the distribution. OC curves
are also drawn.


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References


Chakraborty, A. B. and Bhattacharya, S. K. (1989). CUSUM control charts for doubly truncated geometric and Poisson distributions. pp. 509-512. Proceedings of Quality for Progress and Development, Asian Congress on Quality and Reliability. Wiley Eastern Limited.

Chakraborty, A. B. and Bhattacharya, S. K. (1991). Cumulative sum control chart for a doubly truncated

binomial distribution. Egyptian Statistical Journal, 35, 119-124.

Chakraborty, A. B. and Kakoty, S. K. (1987). Control charts for ZTPD. Journal of Indian Association for Productivity, Quality and Reliability, 12, 17-25.

Chakraborty, A. B. and Khurshid, A. (2011). One-sided cumulative sum (CUSUM) control charts for the zerotruncated binomial distribution. Economic Quality Control, 26, 41-51.

Chakraborty, A. B. and Khurshid, A. (2012). Control charts for doubly-truncated binomial distributions. Economic Quality Control, 27, 187-194.

Chakraborty, A. B. and Singh,. B. P. (1990). Shewhart control chart for ZTPD. pp. 18-24. Proceedings of National

Seminar on Quality and Reliability NIQR, Trivandrum, India.

Dou, Y. and Ping, S. (2002). One-sided control charts for the mean of positively skewed distributions. Total Quality Management, 13, 1021-1033.

Hoffman, D. (2003). Negative binomial control limits for count data with extra-Poisson variation. Pharmaceutical Statistics, 2, 127-132.

Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, Third Edition. John Wiley, New

York.

Kaminsky, F. C., Benneyan, J. C., Davis, R. D. and Burke, R. J. (1992). Statistical control charts based on a geometric distribution. Journal of Quality Technology, 24, 63-69.

Khurshid, A. and Chakraborty, A. B. (2013). CUSUM control charts for zero-truncated negative binomial and

geometric distributions. Revista Investigación Operacional, 34, 195-204

]Khurshid, A., Ageel, M. I. and Lodhi, R. A. (2005). On

confidence intervals for the negative binomial distribution.

Revista Investigación Operacional, 26, 59-70.

Ma, Y. and Zhang, Y. (1995). Q control charts for negative binomial distribution. Computers and Industrial

Engineering, 31, 813-816.

Montgomery, D. C. (2013). Introduction to Statistical Quality Control, Seventh Edition. John Wiley, New York.

Mittag, H. J. and Rinne, H. (1993). Statistical Methods of Quality Assurance. Chapman and Hall, New York.

Promislow, S. D. (2011). Fundamentals of Actuarial Mathematics, Second Edition. John Wiley, New York.

Ryan, T. P. (2011). Statistical Methods for Quality Improvement, Third Edition. John Wiley, New York.

Schwertman, N. C. (2005). Designing accurate control charts based on the geometric and negative binomial

distribution. Quality and Reliability Engineering International, 21, 743-756.

Xie, M. and Goh, T. N. (1997). The use of probability limits for process control based on geometric distribution.

International Journal of Quality and Reliability and Management, 16, 64-73.

Wadsworth, H. M., Stephens, K. S. and Godfrey, A. B. (2002). Modern Methods for Quality Control and

Improvement, Second Edition. John Wiley, India Pvt. Ltd.

Wheeler, D. J. and Chambers, D. S. (2010). Understanding Statistical Process Control, Third Edition. SPC Press,

Knoxville, TN.

Woodall, W. H. (1997). Control Charting Based on Attribute Data: Bibliography and Review. Journal of

Quality Technology, 29, 172-183.

Zelterman, D. (2004). Discrete Distributions: Applications in the Health Sciences. John Wiley, New

York




DOI: http://dx.doi.org/10.22555/pjets.v4i1.521

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Chief Editor

Prof. Dr. Tariq Rahim Soomro 
Dean
College of Computer Science & Information Systems

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 College of Computer Science & Information Systems

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Dr. Insia Hussain
Dr. Dr. Ehsan Rehman
Dr. Imran Majid
Dr. Khurram Iqbal
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Editorial Advisory Board (International)

Prof. Dr. Mazliham Mohd Su'ud, President, Universiti Kuala Lumpur, Malaysia
Prof. Dr. Ghassan Al-Qaimari, President, Emirates College of Technology, Abu Dhabi, UAE
Prof. Dr. Patrice Boursier, Universite de La Rochelle, La Rochelle, France
Prof. Dr. Mudassir Uddin, Professor, University of Karachi, Pakistan
Dr. Nadeem Doudpota, Associate Professor, Al-Baha University, KSA
Dr. Haithem Abdelrazaq Almefleh, Associate Professor, Yarmouk University, Yarmouk, Jordan
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