# Median Control Chart using the Bootstrap Method

• Lim, Soo-Duck (Department of Information and Statistics, Chungbuk National University) ;
• Park, Hyo-Il (Department of Statistics, Chong-Ju University) ;
• Cho, Joong-Jae (Department of Information and Statistics, Chungbuk National University)
• Published : 2007.08.31

#### Abstract

This research considers to propose the control charts using median for the location parameter. In order to decide the control limits, we apply several bootstrap methods through the approach obtaining the confidence interval except the standard bootstrap method. Then we illustrate our procedure using an example and compare the performance among the various bootstrap methods by obtaining the length between control limits through the simulation study. The standard bootstrap may be apt to yield shortest length while the bootstrap-t method, the longest one. Finally we comment briefly about some specific features as concluding remarks.

#### References

1. Alloway, J. A. Jr. and Raghavachari, M. (1991). Control chart based on the HodgesLehmann estimator. Journal of Quality Technology, 23, 336-347 https://doi.org/10.1080/00224065.1991.11979350
2. Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7, 1-26 https://doi.org/10.1214/aos/1176344552
3. Grimshaw, S. D. and Alt, F. B. (1997). Control charts for quantile function values. Journal of Quality Technology, 29, 1-7 https://doi.org/10.1080/00224065.1997.11979719
4. Gunter, B. H. (1989). The use and abuse of $C_{pk}$, Part 2, Quality Progress, 22, 108-109
5. Khoo, M. B. C. (2005). A control chart based on sample median for the detection of a permanent shift in the process mean. Quality Engineering, 17, 243-257 https://doi.org/10.1081/QEN-200057329
6. Liu, R. Y. and Tang, J. (1996). Control charts for dependent and independent measurements based on bootstrap methods. Journal of the American Statistical Association, 91, 1694-1700 https://doi.org/10.2307/2291598
7. Nelson, L. S. (1982). Control chart for medians. Journal of Quality Technology, 14, 226-227 https://doi.org/10.1080/00224065.1982.11978826
8. Seppala, T., Moskowitz, H., Plante, R. and Tang, J. (1995). Statistical process control via the subgroup bootstrap. Journal of Quality Technology, 27, 139-153 https://doi.org/10.1080/00224065.1995.11979577
9. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York
10. Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer-Verlag, New
11. York. Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap. Annals of Statistics, 9, 1187-1195 https://doi.org/10.1214/aos/1176345636
12. Wilks, S. S. (1961). Mathematical Statistics. John Wiley & Sons, New York
13. Jones, L. A. and Woodall, W. H. (1998). The performance of bootstrap control charts. Journal of Quality Technology, 30, 362-375 https://doi.org/10.1080/00224065.1998.11979872
14. 박효일, (2006), 중앙값을 이용한 관리도의 구축. Journal of the Korean Data Analysis Society, 8, 1545-1555
15. Maritz, J. S. and Jarrett, R. G. (1978). A note on estimating the variance of the sample median. Journal of the American Statistical Association, 73, 194-196 https://doi.org/10.2307/2286545
16. Janacek, G. J. and Meikle, S. E. (1997). Control charts based on medians. Journal of the Royal Statistical Society, Ser. D- The Statistician, 46, 19-3l https://doi.org/10.1111/1467-9884.00056