One-Sided Cumulative Sum (CUSUM) Control Charts for the Erlang-Truncated Exponential Distribution
Department of Statistics, School of Mathematical Sciences, University of Dodoma
Dodoma, Tanzania, PO Box:259
E-mail: gaddesrao@yahoo.com
Received:
Received: 13 September 2013; revised: 15 October 2013; accepted: 17 October 2013; published online: 10 December 2013
DOI: 10.12921/cmst.2013.19.04.229-234
OAI: oai:lib.psnc.pl:459
Abstract:
In this article, we construct one-sided cumulative sum (CUSUM) control charts for controlling the parameters of a random variable with erlang-truncated exponential distribution. The rejection of the Wald’s sequential probability ratio test (SPRT) is viewed as the decision lines of a CUSUM control chart for which the variate is a quality characteristic. Parameters of the CUSUM chart, e.g. lead distance and mask angle, are presented. The results show that the Average Run Length (ARL) of the resulting control charts changes substantially for a slight shift in the parameters of the distribution.
Key words:
Average Run Length (ARL), Cumulative Sum (CUSUM) Control Chart, erlang-truncated exponential distribution, Sequential Probability Ratio Test (SPRT)
References:
[1] A.B. Chakraborty, A. Khurshid, One-Sided Cumulative Sum
(CUSUM) Control Charts for the Zero-Truncated Binomial
Distribution, Economic Quality Control 26, 41-51 (2011).
[2] A.R. El-Alosey, Random sum of new type of mixture of dis-
tribution, International Journal of Statistics and Systems 2,
49-57 (2007).
[3] N.L. Johnson, A simple theoretical approach to Cumulative
sum control charts, Journal of Amer. Statist. Assoc. 56, 835-
840 (1961).
[4] N. L. Johnson, Cumulative sum control charts and the Weibull
distribution, Technometrics 8 (3), 481-491 (1966).
[5] N.L. Johnson, F.C. Leone, Cumulative sum control charts:
Mathematical principles applied to their construction and
use, Indust. Qual. Control 19, 22-28 (1962).
[6] R.R.L. Kantam, G.S. Rao, Cumulative Sum Control Chart
for log-logistic distribution, InterStat, online Journal, July,
1-9 (2006).
[7] J.M. Lucas, Combined Shewhart-CUSUM quality control
schemes, Journal of Quality Technology 14, 51-59 (1982).
[8] M. Mohsin, Recurrence relation for single and product mo-
ments of record values from Erlang-truncated exponential dis-
tribution, World Applied Science Journal, 6, 279-282 (2009).
[9] M. Mohsin, S. Shahbaz, M.Q. Shahbaz, A characterization
of Erlang-truncated exponential distribution in record values
and its use in mean residual life, Pakistan Journal of Statistics
and Operations Research 6(2), 143-148 (2010).
[10] D.C. Montgomery, Introduction to Statistical Quality Control,
Third edition, John Wiley&Sons, New York 2001.
[11] S.P. Nabar, S. Bilgi, Cumulative sum control chart for the
Inverse Gaussian distribution, Journal of Indian Statistical
Association 32, 9-14 (1994).
[12] E.S. Page, Continuous inspection schemes, Biometrika 41,
100-115 (1954).
[13] E.S. Page, Cumulative sum charts, Technometrics 3, 1-9
(1961).
[14] A. Wald, Sequential analysis, John Wiley&Sons, New York
1947.
In this article, we construct one-sided cumulative sum (CUSUM) control charts for controlling the parameters of a random variable with erlang-truncated exponential distribution. The rejection of the Wald’s sequential probability ratio test (SPRT) is viewed as the decision lines of a CUSUM control chart for which the variate is a quality characteristic. Parameters of the CUSUM chart, e.g. lead distance and mask angle, are presented. The results show that the Average Run Length (ARL) of the resulting control charts changes substantially for a slight shift in the parameters of the distribution.
Key words:
Average Run Length (ARL), Cumulative Sum (CUSUM) Control Chart, erlang-truncated exponential distribution, Sequential Probability Ratio Test (SPRT)
References:
[1] A.B. Chakraborty, A. Khurshid, One-Sided Cumulative Sum
(CUSUM) Control Charts for the Zero-Truncated Binomial
Distribution, Economic Quality Control 26, 41-51 (2011).
[2] A.R. El-Alosey, Random sum of new type of mixture of dis-
tribution, International Journal of Statistics and Systems 2,
49-57 (2007).
[3] N.L. Johnson, A simple theoretical approach to Cumulative
sum control charts, Journal of Amer. Statist. Assoc. 56, 835-
840 (1961).
[4] N. L. Johnson, Cumulative sum control charts and the Weibull
distribution, Technometrics 8 (3), 481-491 (1966).
[5] N.L. Johnson, F.C. Leone, Cumulative sum control charts:
Mathematical principles applied to their construction and
use, Indust. Qual. Control 19, 22-28 (1962).
[6] R.R.L. Kantam, G.S. Rao, Cumulative Sum Control Chart
for log-logistic distribution, InterStat, online Journal, July,
1-9 (2006).
[7] J.M. Lucas, Combined Shewhart-CUSUM quality control
schemes, Journal of Quality Technology 14, 51-59 (1982).
[8] M. Mohsin, Recurrence relation for single and product mo-
ments of record values from Erlang-truncated exponential dis-
tribution, World Applied Science Journal, 6, 279-282 (2009).
[9] M. Mohsin, S. Shahbaz, M.Q. Shahbaz, A characterization
of Erlang-truncated exponential distribution in record values
and its use in mean residual life, Pakistan Journal of Statistics
and Operations Research 6(2), 143-148 (2010).
[10] D.C. Montgomery, Introduction to Statistical Quality Control,
Third edition, John Wiley&Sons, New York 2001.
[11] S.P. Nabar, S. Bilgi, Cumulative sum control chart for the
Inverse Gaussian distribution, Journal of Indian Statistical
Association 32, 9-14 (1994).
[12] E.S. Page, Continuous inspection schemes, Biometrika 41,
100-115 (1954).
[13] E.S. Page, Cumulative sum charts, Technometrics 3, 1-9
(1961).
[14] A. Wald, Sequential analysis, John Wiley&Sons, New York
1947.
