Steady-state behavior of nonparametric control charts using sign statistic
Khilare, Shashikant Kuber; Shirke, Digambar Tukaram
http://dx.doi.org/10.1590/0103-6513.113112
Production, vol.25, n4, p.739-749, 2015
Abstract
If process is running for a long period in an in-control condition, it will reach in a steady-state condition. In order to study the long term properties of a control chart, it is appropriate to investigate the steady-state average time to signal. In this article, we discussed runs rules representation of a nonparametric synthetic control chart using sign statistic for detecting shifts in location parameter. We compared zero-state average time to signal with steady-state average time to signal of the synthetic control chart for symmetric and asymmetric distributions. We also present the m-of-m control chart using sign statistic. For comparison study, we computed average time to signal of the m-of-m control chart, the sign chart (1-of-1 chart) and the synthetic control chart for normal, Cauchy, double exponential and gamma distributions. Steady-state and zero-state performance of the m-of-m control chart with m = 2, 3 compared with the sign chart (1-of-1 chart) and synthetic control chart. The zero-state and steady-state average time to signal of the synthetic and the m-of-m control charts computed using Markov chain approach.
Keywords
Steady-state. Markov chain. Synthetic. Nonparametric. Average time to signal.
References
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Amin, R. W., Reynolds Junior, M. R., & Bakir, S. T. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistic: Theory and Methods, 24(6), 1597-1623. http://dx.doi.org/10.1080/03610929508831574
Bakir, S. T. (2004). A distribution-free shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623. http://dx.doi.org/10.1081/QEN-120038022
Bakir, S. T. (2006). Distribution-free quality control charts based on signed-rank-like statistic. Communications in Statistics: Theory and Methods, 35(4), 743-757. http://dx.doi.org/10.1080/03610920500498907
Bakir, S. T., & Reynolds Junior, M. R. (1979). A nonparametric procedure for process control based on within-group ranking. Technometrics, 21(2), 175-183. http://dx.doi.org/10.1080/00401706.1979.10489747
Bourke, P. D. (1991). Detecting a shift in fraction nonconforming using run length control chart with 100% inspection. Journal of Quality Technology, 3(2), 51-68.
Chakraborti, S., & Eryilmaz, S. (2007). A nonparametric shewhart-type signed-rank control chart based on runs. Communications in Statistic, 36(2), 335-356. http://dx.doi.org/10.1080/03610910601158427
Chakraborti, S., & Van de Wiel, M. A. (2008). A nonparametric control charts based on mann-whitney statistic. IMS Collection, 1, 156-172. http://dx.doi.org/10.1214/193940307000000112
Champ, W. C. (1992). Steady-state run length analysis of a shewhart control chart with supplementary runs rules. Communications in Statistics: Theory and Methods, 21(3), 765-777. http://dx.doi. org/10.1080/03610929208830813
Crosier, R. B. (1986). A new two-sided cumulative sum quality control scheme. Technometrics, 28(3), 187-194. http://dx.doi.org/10.1080/00401706.1986.10488126
Davis, R. B., & Woodall, W. H. (2002). Evaluating and improving the synthetic control chart. Journal of Quality Technology, 34(2), 200-208.
Ho, L. L., & Costa, A. F. B. (2011). Monitoring a wandering mean with an np chart. Produção, 21(2), 254-258. http://dx.doi.org/10.1590/S0103-65132011005000027
Human, S. W., Chakraborti, S., & Smit, C. F. (2010). Nonparametric shewhart-type sign control charts based on runs. Communications in Statistics: Theory and Methods, 39(11), 2046-2062. http://dx.doi.org/10.1080/03610920902969018
Khilare, S. K., & Shirke, D. T. (2010). A nonparametric synthetic control chart using sign statistic. Communications in 749 Statistics: Theory and Methods, 39(18), 3282-3293. http://dx.doi.org/10.1080/03610920903249576
Lim, T., & Cho, M. (2009). Design of control charts with m-of-m runs rules. Quality and Reliability Engineering International, 25(8), 1085-1101. http://dx.doi.org/10.1002/qre.1023
McGilchrist, C. A, & Woodyer, K. D. (1975). Note on a distribution-free CUSUM technique. Technometrics, 17(3), 321-325. http://dx.doi.org/10.10 80/00401706.1975.10489335
Saccucci, M. S., & Lucas, J. M. (1990). Average run length for exponentially weighted moving average control schemes using the markov chain approach. Journal of Quality Technology, 22(2), 154-162.
Wu, Z., & Spedding, T. A. (2000). A synthetic control chart for detecting small shifts in the process mean. Journal of Quality Technology, 32, 32-38.
Wu, Z., Yeo, S. H., & Spedding, T. A. (2001). A synthetic control chart for detecting fraction nonconforming increases. Journal of Quality Technology, 33(1), 104-111.
Amin, R. W., Reynolds Junior, M. R., & Bakir, S. T. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistic: Theory and Methods, 24(6), 1597-1623. http://dx.doi.org/10.1080/03610929508831574
Bakir, S. T. (2004). A distribution-free shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623. http://dx.doi.org/10.1081/QEN-120038022
Bakir, S. T. (2006). Distribution-free quality control charts based on signed-rank-like statistic. Communications in Statistics: Theory and Methods, 35(4), 743-757. http://dx.doi.org/10.1080/03610920500498907
Bakir, S. T., & Reynolds Junior, M. R. (1979). A nonparametric procedure for process control based on within-group ranking. Technometrics, 21(2), 175-183. http://dx.doi.org/10.1080/00401706.1979.10489747
Bourke, P. D. (1991). Detecting a shift in fraction nonconforming using run length control chart with 100% inspection. Journal of Quality Technology, 3(2), 51-68.
Chakraborti, S., & Eryilmaz, S. (2007). A nonparametric shewhart-type signed-rank control chart based on runs. Communications in Statistic, 36(2), 335-356. http://dx.doi.org/10.1080/03610910601158427
Chakraborti, S., & Van de Wiel, M. A. (2008). A nonparametric control charts based on mann-whitney statistic. IMS Collection, 1, 156-172. http://dx.doi.org/10.1214/193940307000000112
Champ, W. C. (1992). Steady-state run length analysis of a shewhart control chart with supplementary runs rules. Communications in Statistics: Theory and Methods, 21(3), 765-777. http://dx.doi. org/10.1080/03610929208830813
Crosier, R. B. (1986). A new two-sided cumulative sum quality control scheme. Technometrics, 28(3), 187-194. http://dx.doi.org/10.1080/00401706.1986.10488126
Davis, R. B., & Woodall, W. H. (2002). Evaluating and improving the synthetic control chart. Journal of Quality Technology, 34(2), 200-208.
Ho, L. L., & Costa, A. F. B. (2011). Monitoring a wandering mean with an np chart. Produção, 21(2), 254-258. http://dx.doi.org/10.1590/S0103-65132011005000027
Human, S. W., Chakraborti, S., & Smit, C. F. (2010). Nonparametric shewhart-type sign control charts based on runs. Communications in Statistics: Theory and Methods, 39(11), 2046-2062. http://dx.doi.org/10.1080/03610920902969018
Khilare, S. K., & Shirke, D. T. (2010). A nonparametric synthetic control chart using sign statistic. Communications in 749 Statistics: Theory and Methods, 39(18), 3282-3293. http://dx.doi.org/10.1080/03610920903249576
Lim, T., & Cho, M. (2009). Design of control charts with m-of-m runs rules. Quality and Reliability Engineering International, 25(8), 1085-1101. http://dx.doi.org/10.1002/qre.1023
McGilchrist, C. A, & Woodyer, K. D. (1975). Note on a distribution-free CUSUM technique. Technometrics, 17(3), 321-325. http://dx.doi.org/10.10 80/00401706.1975.10489335
Saccucci, M. S., & Lucas, J. M. (1990). Average run length for exponentially weighted moving average control schemes using the markov chain approach. Journal of Quality Technology, 22(2), 154-162.
Wu, Z., & Spedding, T. A. (2000). A synthetic control chart for detecting small shifts in the process mean. Journal of Quality Technology, 32, 32-38.
Wu, Z., Yeo, S. H., & Spedding, T. A. (2001). A synthetic control chart for detecting fraction nonconforming increases. Journal of Quality Technology, 33(1), 104-111.
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