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Monitoring the asymmetry parameter of a skew-normal distribution

  • Hyun Jun Kim (Department of Applied Statistics, Chung-Ang University) ;
  • Jaeheon Lee (Department of Applied Statistics, Chung-Ang University)
  • 투고 : 2023.07.31
  • 심사 : 2023.11.27
  • 발행 : 2024.01.31

초록

In various industries, especially manufacturing and chemical industries, it is often observed that the distribution of a specific process, initially having followed a normal distribution, becomes skewed as a result of unexpected causes. That is, a process deviates from a normal distribution and becomes a skewed distribution. The skew-normal (SN) distribution is one of the most employed models to characterize such processes. The shape of this distribution is determined by the asymmetry parameter. When this parameter is set to zero, the distribution is equal to the normal distribution. Moreover, when there is a shift in the asymmetry parameter, the mean and variance of a SN distribution shift accordingly. In this paper, we propose procedures for monitoring the asymmetry parameter, based on the statistic derived from the noncentral t-distribution. After applying the statistic to Shewhart and the exponentially weighted moving average (EWMA) charts, we evaluate the performance of the proposed procedures and compare it with previously studied procedures based on other skewness statistics.

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참고문헌

  1. Arellano-Valle RB, Gomez HW, and Quintana FA (2004). A new class of skew-normal distributions, Communications in Statistics-Theory and Methods, 33, 1465-1480. https://doi.org/10.1081/STA-120037254
  2. Azzalini A (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.
  3. Azzalini A (1986). Further results on a class of distributions which includes the normal ones, Statistica, 46, 199-208.
  4. Azzalini A (2005). The skew-normal distribution and related multivariate families, Scandinavian Journal of Statistics, 32, 159-188. https://doi.org/10.1111/j.1467-9469.2005.00426.x
  5. Chen JT, Gupta AK, and Nguyen TT (2004). The density of the skew normal sample mean and its applications, Journal of Statistical Computation and Simulation, 74, 487-494. https://doi.org/10.1080/0094965031000147687
  6. Figueiredo F and Gomes MI (2013). The skew-normal distribution in SPC, REVSTAT-Statistical Journal, 11, 83-104.
  7. Guo B and Wang BX (2015). The design of the ARL-unbiased S2 chart when the in-control variance is estimated, Quality and Reliability Engineering International, 31, 501-511. https://doi.org/10.1002/qre.1608
  8. Hawkins DM and Deng Q (2009). Combined charts for mean and variance information, Journal of Quality Technology, 41, 415-425. https://doi.org/10.1080/00224065.2009.11917795
  9. Henze N (1986). A probabilistic representation of the 'skew-normal' distribution, Scandinavian Journal of Statistics, 13, 271-275.
  10. Knoth S (2015). Run length quantiles of EWMA control charts monitoring normal mean or/and variance, International Journal of Production Research, 53, 4629-4647. https://doi.org/10.1080/00207543.2015.1005253
  11. Li C, Mukherjee A, Su Q, and Xie M (2016). Design and implementation of two CUSUM schemes for simultaneously monitoring the process mean and variance with unknown parameters, Quality and Reliability Engineering International, 32, 2961-2975. https://doi.org/10.1002/qre.1980
  12. Li C, Mukherjee A, Su Q, and Xie M (2019). Some monitoring procedures related to asymmetry parameter of Azzalini's skew-normal model, REVSTAT-Statistical Journal, 17, 1-24.
  13. Li Z, Zou C, Wang Z, and Huwang L (2013). A multivariate sign chart for monitoring process shape parameters, Journal of Quality Technology, 45, 149-165. https://doi.org/10.1080/00224065.2013.11917923
  14. Mameli V and Musio M (2013). A generalization of the skew-normal distribution: The beta skewnormal, Communications in Statistics-Theory and Methods, 42, 2229-2244. https://doi.org/10.1080/03610926.2011.607530
  15. McCracken AK, Chakraborti S, and Mukherjee A (2013). Control charts for simultaneous monitoring of unknown mean and variance of normally distributed processes, Journal of Quality Technology, 45, 360-376. https://doi.org/10.1080/00224065.2013.11917944
  16. Mukherjee A, Abd-Elfattah AM, and Pukait B (2013). A rule of thumb for testing symmetry about an unknown median against a long right tail, Journal of Statistical Computation and Simulation, 84, 2138-2155. https://doi.org/10.1080/00949655.2013.784316
  17. Peng Y, Xu L, and Reynolds MR Jr (2015). The design of the variable sampling interval generalized likelihood ratio chart for monitoring the process mean, Quality and Reliability Engineering International, 31, 291-296. https://doi.org/10.1002/qre.1587
  18. Rahman S and Hossain F (2008). A forensic look at groundwater arsenic contamination in Bangladesh, Environ Forensics, 9, 364-374. https://doi.org/10.1080/15275920801888400
  19. Ross GJ and Adams NM (2012). Two nonparametric control charts for detecting arbitrary distribution changes, Journal of Quality Technology, 44, 102-116. https://doi.org/10.1080/00224065.2012.11917887
  20. Ryu JH, Wan H, and Kim S (2010). Optimal design of a CUSUM chart for a mean shift of unknown size, Journal of Quality Technology, 42, 311-326. https://doi.org/10.1080/00224065.2010.11917826
  21. Sheu SH, Huang CJ, and Hsu TS (2012). Extended maximum generally weighted moving average control chart for monitoring process mean and variability, Computers and Industrial Engineering, 62, 216-225. https://doi.org/10.1016/j.cie.2011.09.009
  22. Shu L, Yeung HF, and Jiang W (2010). An adaptive CUSUM procedure for signaling process variance changes of unknown sizes, Journal of Quality Technology, 42, 69-85. https://doi.org/10.1080/00224065.2010.11917807
  23. Vincent R and Walsh TD (1997). Quantitative measurement of symmetry in CBED patterns, Ultramicroscopy, 70, 83-94. https://doi.org/10.1016/S0304-3991(97)00080-6
  24. Wu Z, Yang M, Khoo MBC, and Yu FJ (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. https://doi.org/10.1016/j.ejor.2009.12.005
  25. Zou C and Tsung F (2010). Likelihood ratio-based distribution-free EWMA control charts, Journal of Quality Technology, 42, 174-196.  https://doi.org/10.1080/00224065.2010.11917815