Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T17:24:53.110Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERAL CLASS OF CUSUM STATISTICS

Published online by Cambridge University Press:  23 March 2015

Christopher S. Withers  and
Saralees Nadarajah
Christopher S. Withers
Affiliation:
Industrial Research Limited, Lower Hutt, New Zealand E-mail: [email protected]
Saralees Nadarajah
Affiliation:
University of Manchester, Manchester M13 9PL, UK E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Much of the work relating to control charts is on monitoring observations for changes in mean. Here, for the first time, we develop a class of cusum-type statistics to monitor any aspect of a process (for example, mean, variance, skewness or kurtosis). This development is based on recently published papers by the authors.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 3 , July 2015 , pp. 361 - 384
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Abramowitz, M. & Stegun, I.A. (1964). Handbook of mathematical functions. United States Department of Commerce, National Bureau of Standards, Applied Mathematics Series, vol. 55.Google Scholar
2. Al-Tubi, M.A.S. & Al-Raheem, K.F. (2010). Rolling element bearing faults detection, A time-domain analysis. Caledonian Journal of Engineering 6: 44–48.Google Scholar
3. Andreou, E. & Ghysels, E. (2003). Tests for breaks in the conditional co-movements of asset returns. Statistica Sinica 13: 1045–1073.Google Scholar
4. Aue, A., Hörmann, S., Horváth, L. & Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models. Annals of Statistics 37: 4046–4087.Google Scholar
5. Caplier, A. (2001). Lip detection and tracking. In: Proceedings of the 11th International Conference on Image Analysis and Processing, volume 4393, pp. 22–29.Google Scholar
6. Cascon, A., Keating, C. & Shadwick, W.F. (2003). The Omega function. Working paper.Google Scholar
7. Castagliola, P. & Maravelakis, P.E. (2011). A CUSUM control chart for monitoring the variance when parameters are estimated. Journal of Statistical Planning and Inference 141: 1463–1478.Google Scholar
8. Chen, J. & Gupta, A.K. (1997). Testing and locating variance changepoints with application to stock prices. Journal of the American Statistical Association 92: 739–747.Google Scholar
9. Chinnam, R.B., Rai, B. & Singh, N. (2004). Tool-condition monitoring from degradation signals using Mahalanobis–Taguchi system analysis. In: Robust Engineering pp. 343–351.Google Scholar
10. Csörgo, M. & Révész, P. (1981). Strong approximations in probability and statistics. New York: Academic Press.Google Scholar
11. Cushey, M.A., Bellin, A. & Rubin, Y. (1995). Three-dimensional modelling of flow and reactive transport in heterogeneous porous media. In: Groundwater Quality—Remediation and Protection, IAHS Publication no. 225, pp. 257–261.Google Scholar
12. Davis, R.A., Lee, T.C.M. & Rodriguez-Yam, G.A. (2008). Break detection for a class of nonlinear time series models. Journal of Time Series Analysis 29: 834–867.Google Scholar
13. Lee, S., Na, O. & Na, S. (2003). On the cusum of squares test for variance change in nonstationary and nonparametric time series models. Annals of the Institute of Statistical Mathematics 55: 467–485.Google Scholar
14. Montgomery, D.C. (1991). Introduction to statistical quality control. 2nd ed. New York: John Wiley and Sons.Google Scholar
15. Neuberger, A. (2011). Realized skewness. Coventry, UK: Warwick Business School, Warwick University.Google Scholar
16. Park, S., Lee, S. & Jeon, J. (2000). The cusum of squares test for variance changes in infinite order autoregressive models. Journal of the Korean Statistical Society 29: 351–360.Google Scholar
17. Rempala, G. & Wesolowski, J. (2002). Asymptotics for products of sums and U-statistics. Electronic Communications in Probability 7: 47–54.Google Scholar
18. Reynolds, M.R. & Stoumbos, Z.G. (2010). Robust CUSUM charts for monitoring the process mean and variance. Quality and Reliability Engineering International 26: 453–473.Google Scholar
19. Rickett, B.J. (1975). Disturbances in the solar wind from IPS measurements in August 1972. Solar Physics 43: 237–247.Google Scholar
20. Shu, L., Yeung, H.-F. & Jiang, W. (2010). An adaptive CUSUM procedure for signaling process variance changes of unknown sizes. Journal of Quality Technology 42: 69–85.Google Scholar
21. Vazza, F., Roediger, E. & Brüggen, M. (2012). Turbulence in the ICM from mergers, cool-core sloshing and jets: results from a new multi-scale filtering approach.Google Scholar
22. Wang, J., Jia, M., Zhu, L., Yuan, Z., Li, P., Chang, C., Luo, J., Liu, M. & Shi, T. (2010). Systematical detection of significant genes in microarray data by incorporating gene interaction relationship in biological systems. PLoS ONE 5: e13721.Google Scholar
23. Withers, C.S. (1982a). The distribution and quantiles of a function of parameter estimates. Annals of the Institute of Statistical Mathematics, A 34: 55–68.Google Scholar
24. Withers, C.S. (1982b). Second order inference for asymptotically normal random variables. Sankhyā, B 44: 1–9.Google Scholar
25. Withers, C.S. (1983). Expansions for the distribution and quantiles of a regular functional of the empirical distribution with applications to nonparametric confidence intervals. Annals of Statistics 11: 577–587.Google Scholar
26. Withers, C.S. (1984). Asymptotic expansions for distributions and quantiles with power series cumulants. Journal of the Royal Statistical Society B 46: 389–396.Google Scholar
27. Withers, C.S. & Nadarajah, S. (2009). Accurate tests and intervals based on linear cusum statistics. Statistics and Probability Letters 79: 689–697.Google Scholar
28. Withers, C.S. & Nadarajah, S. (2009). Accurate tests and intervals based on non-linear cusum statistics. Statistics and Probability Letters 79: 2242–2250.Google Scholar
29. Withers, C.S. & Nadarajah, S. (2012). Accurate tests and intervals based on multivariate cusum statistics. Sequential Analysis 31: 78–87.Google Scholar
30. Withers, C.S. & Nadarajah, S. (2013). Weighted cusums for increased power near the end points. Methodology and Computing in Applied Probability 15: 379–405.Google Scholar
31. Wu, Z. & Tian, Y. (2005). Weighted-loss-function CUSUM chart for monitoring mean and variance of a production process. International Journal of Production Research 43: 3027–3044.Google Scholar
32. Wu, Z., Yang, M., Khoo, M.B.C. & Yu, F.-J. (2010). Optimization designs and performance comparison of two CUSUM schemes for monitoring process shifts in mean and variance. European Journal of Operational Research 205: 136–150.Google Scholar
33. Wu, Z., Zhang, S. & Wang, P. (2007). A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance. Quality and Reliability Engineering International 23: 157–170.Google Scholar
34. Zhang, S. & Wu, Z. (2006). Monitoring the process mean and variance using a weighted loss function CUSUM scheme with variable sampling intervals. IIE Transactions 38: 377–387.Google Scholar