DOI QR코드

DOI QR Code

A Note on the Robustness of the X Chart to Non-Normality

  • Lee, Sung-Im (Department of Statistics, Dankook University)
  • Received : 2012.06.18
  • Accepted : 2012.07.21
  • Published : 2012.09.30

Abstract

These days the interest of quality leads to the necessity of control charts for monitoring the process in various fields of practical applications. The $\overline{X}$ chart is one of the most widely used tools for quality control that also performs well under the normality of quality characteristics. However, quality characteristics tend to have nonnormal properties in real applications. Numerous recent studies have tried to find and explore the performance of $\overline{X}$ chart due to non-normality; however previous studies numerically examined the effects of non-normality and did not provide any theoretical justification. Moreover, numerical studies are restricted to specific type of distributions such as Burr or gamma distribution that are known to be flexible but can hardly replace other general distributions. In this paper, we approximate the false alarm rate(FAR) of the $\overline{X}$ chart using the Edgeworth expansion up to 1/n-order with the fourth cumulant. This allows us to examine the theoretical effects of nonnormality, as measured by the skewness and kurtosis, on $\overline{X}$ chart. In addition, we investigate the effect of skewness and kurtosis on $\overline{X}$ chart in numerical studies. We use a skewed-normal distribution with a skew parameter to comprehensively investigate the effect of skewness.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. Azzalini, A. (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.
  2. Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones, Statistics, 44, 199-208.
  3. Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skewnormal distri- butions, Journal of the Royal Statistical Society, Series B, 61, 579-602. https://doi.org/10.1111/1467-9868.00194
  4. Azzalini, A. and Dalla-Valle, A. (1996). The multivariate skew-normal distribution, Biometrika, 83, 715-726. https://doi.org/10.1093/biomet/83.4.715
  5. Bai, D. S. and Choi, I. S. (1995). $\bar{X}$ and R control charts for skewed populations, Journal of Quality Technology, 27, 120-131.
  6. Burr, I. W. (1967) The effect of non-normality on constants for $\bar{X}$ and R charts, Industrial Quality Control, 24, 563-569.
  7. Burrows, P. M. (1962). $\bar{X}$ control schemes for a production variable with skewed distribution, Statistician, 12, 296-312. https://doi.org/10.2307/2986950
  8. Chakraborti, S., van der Laan, P. and van de Wiel, M. A. (2004). A class of distribution-free control charts, Applied Statistics, 53, 443-462. https://doi.org/10.1111/j.1467-9876.2004.0d489.x
  9. Chan, L. K., Hapuarachchi, K. P. and Macpherson, B. D. (1988). Robustness of X and R charts, IEEE. Transactions on Reliability, 37, 117-123. https://doi.org/10.1109/24.3728
  10. Janacek, G. J. and Meikle, S. E. (1997). Control charts based on medians, The Statistician, 46, 19-31. https://doi.org/10.1111/1467-9884.00056
  11. Roes, K. C. B. and Does, R. J. M. M. (1995). Shewhart-type charts in nonstandard situations(with discussions), Technometrics, 37, 15-40. https://doi.org/10.1080/00401706.1995.10485882
  12. Schilling, E. G. and Nelson, P. R. (1976). The effect of non-normality on the control limits of X chart, Journal of Quality Technology, 8, 183-188.
  13. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand, New York.
  14. Yourstone, S. A. and Zimmer, W. J. (1992). Non-normality and the design of control charts for averages, Decision Sciences, 23, 1099-1113. https://doi.org/10.1111/j.1540-5915.1992.tb00437.x
  15. Wheeler, D. J. and Chambers, D. S. (1992). Understanding Statistical Process Control, SPC Press, Knoxville, TN.