A CUSUM control chart for monitoring the variance when parameters are estimated. (English) Zbl 1204.62189
Summary: The CUSUM control chart has been widely used for monitoring the process variance. It is usually used assuming that the nominal process variance is known. However, several researchers have shown that the ability of control charts to signal when a process is out of control is seriously affected unless the process parameters are estimated from a large in-control Phase I data set. We derive the run length properties of a CUSUM chart for monitoring dispersion with estimated process variance and we evaluate the performance of this chart by comparing it with the same chart but with assumed known process parameters.
MSC:
| 62P30 | Applications of statistics in engineering and industry; control charts |
References:
| [1] | Albers, W.; Kallenberg, W., Estimation in shewhart control charts: effects and corrections, Metrika, 59, 3, 207-234 (2004) · Zbl 1147.62397 |
| [2] | Albers, W.; Kallenberg, W.; Nurdiati, S., Parametric control charts, Journal of Statistical Planning and Inference, 124, 1, 159-184 (2004) · Zbl 1095.62132 |
| [3] | Brook, D.; Evans, D., An approach to the probability distribution of CUSUM run length, Biometrika, 59, 3, 539-549 (1972) · Zbl 0265.62038 |
| [4] | Castagliola, P.; Celano, G.; Chen, G., The exact run length distribution and design of the \(s^2\) chart when the in-control variance is estimated, International Journal of Reliability Quality and Safety Engineering, 16, 1, 23-38 (2009) |
| [5] | Castagliola, P.; Celano, G.; Fichera, S., Monitoring process variability using EWMA, (Springer Handbook of Engineering Statistics (2006), Springer), 291-325 |
| [6] | Chang, T.; Gan, F., A cumulative sum control chart for monitoring process variance, Journal of Quality Technology, 2, 109-119 (1995) |
| [7] | Chen, G., The mean and standard deviation of the run length distribution of \(\overline{X}\) charts when control limits are estimated, Statistica Sinica, 7, 789-798 (1997) · Zbl 0876.62086 |
| [8] | Chen, G., The run length distributions of the \(R, s\) and \(s^2\) control charts when \(\sigma\) is estimated, The Canadian Journal of Statistics, 26, 311-322 (1998) · Zbl 0914.62078 |
| [9] | Jensen, W.; Jones-Farmer, L.; Champ, C.; Woodall, W., Effects of parameter estimation on control chart properties: a literature review, Journal of Quality Technology, 38, 349-364 (2006) |
| [10] | Jones, L.; Champ, C.; Rigdon, S., Distribution of the CUSUM with estimated parameters, Journal of Quality Technology, 36, 95-108 (2004) |
| [11] | Latouche, G.; Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic Modelling (1999), ASA SIAM · Zbl 0922.60001 |
| [12] | Lucas, J.; Crosier, R., Fast initial response for CUSUM quality control schemes, Technometrics, 24, 199-205 (1982) |
| [13] | Maravelakis, P.; Castagliola, P., An EWMA chart for monitoring the process standard deviation when parameters are estimated, Computational & Statistics Data Analysis, 53, 7, 2653-2664 (2009) · Zbl 1453.62150 |
| [14] | Neuts, M., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (1981), Dover Publications Inc · Zbl 0469.60002 |
| [15] | Page, E., Continuous inspection schemes, Biometrika, 41, 100-114 (1954) · Zbl 0056.38002 |
| [16] | Page, E., Controlling the standard deviation by CUSUM and warning lines, Technometrics, 5, 307-315 (1963) |
| [17] | Srivastava, M., Cusum procedure for monitoring variability, Communications in Statistics—Theory and Methods, 26, 12, 2905-2926 (1997) · Zbl 0954.62570 |
| [18] | Tuprah, K.; Ncube, M., A comparison of dispersion quality control charts, Sequential Analysis, 6, 155-163 (1987) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
