Abstract
The cumulative sum (CUSUM) charting procedure is an important online monitoring procedure which is especially effective for detecting small and moderate shifts. In the design and implementation of an optimal CUSUM chart, the probability density function of an in-control process distribution is assumed to be known. If the density is not known or cannot be approximated using a known density, an optimal CUSUM chart cannot be implemented. We propose a CUSUM chart which does not require the density to be known. Kernel density estimation method will be used to estimate the density of an in-control process distribution. The performance of this chart is investigated for unimodal distributions. The results obtained reveal that this chart works well if we can obtain sufficient observations from an in-control process for kernel density and average run length estimations. An example is given to illustrate the design and implementation of this chart.
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References
Abramowitz, M., & Stegun, I. A. (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publications.
Cencov, N. N. (1962). Evaluation of an unknown density from observations. Soviet Mathematics, 3, 1559–1562.
Duda, R. O., Stork, D. G., & Hart, P. E. (2000). Pattern classification. New York: Wiley.
Gan, F. F. (1991). An optimal design of CUSUM quality control charts. Journal of Quality Technology, 23, 279–286.
Gan, F. F. (1994). Design of optimal exponential CUSUM control charts. Journal of Quality Technology, 26, 109–124.
Gandy, A., & Kvaløy, J. T. (2013). Guaranteed conditional performance of control charts via bootstrap methods. Scandinavian Journal of Statistics. doi:10.1002/sjos.12006.
Goel, A. L., & Wu, S. M. (1971). Determination of A.R.L. and a contour nomogram for CUSUM charts to control normal mean. Technometrics, 13, 221–230.
Hawkins, D. M., & Olwell, D. H. (1997). Inverse Gaussian cumulative sum control charts for location and shape. Journal of the Royal Statistical Society: Series D (The Statistician), 46, 323–335.
Hawkins, D. M., & Olwell, D. H. (1998). Cumulative sum charts and charting for quality improvement. New York: Springer.
Huang, W., Shu, L., Jiang, W., & Tsui, K. -L. (2013). Evaluation of run-length distribution for CUSUM charts under gamma distributions. IIE Transaction, 45, 981–994.
Knoth, S. (2005). Accurate ARL computation for EWMA-S 2 control charts. Statistics and Computing, 15, 341–352.
Knoth, S. (2006). Computation of the ARL for CUSUM-S 2 schemes. Computational Statistics & Data Analysis, 51, 499–512.
Lehmann, E. L. (1990). Model specifiction: the views of Fisher and Neyman, and later developments. Statistical Science, 5, 160–168.
Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distribution. The Annals of Statistics, 14, 1379–1387.
Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41, 100–115.
Page, E. S. (1961). Cumulative sum charts. Technometrics, 3, 1–9.
Parzen, E. (1962). On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33, 1065–1076.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistic, 27, 832–837.
Scott, D. W. (1992). Multivariate density estimation. New York: Wiley-Interscience.
Shewhart, W. A. (1931). Economic control of quality of manufactured product. New York: D. Van Nostrand.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman and Hall.
Vardeman, S., & Ray, D. (1985). Average run lengths for CUSUM schemes when observations are exponentially distributed. Technometrics, 27, 145–150.
Wald, A. (1943). Sequential analysis of statistical data: Theory. A Report Submitted by the Statistical Research Group, Columbia University to the Applied Mathematics Panel, National Defense Research Committee, Sept 1943.
Wald, A. (1945). Sequential tests of statistical hypotheses. The Annals of Mathematical Statistics, 16, 117–186.
Wald, A. (1947). Sequential analysis. New York: Wiley.
Wald, A., & Wolfowitz, J. (1948). Optimum character of the sequential probability ratio test. The Annals of Mathematical Statistics, 19, 326–339.
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The second and third authors are supported by the Academic Research Fund Tier 1 (R-155-000-137-112), Ministry of Education, Singapore.
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Su, J.Y., Gan, F.F., Tang, X. (2015). Optimal Cumulative Sum Charting Procedures Based on Kernel Densities. In: Knoth, S., Schmid, W. (eds) Frontiers in Statistical Quality Control 11. Frontiers in Statistical Quality Control. Springer, Cham. https://doi.org/10.1007/978-3-319-12355-4_8
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DOI: https://doi.org/10.1007/978-3-319-12355-4_8
Publisher Name: Springer, Cham
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