Control charts, widely accepted and applied in industry, are one of powerful techniques of statistical process control. They are used to determine whether a process is in statistical control state and to provide information in diagnosis and process capability. The most popular method to monitor the process is Shewhart X and R control charts proposed by Shewhart (1931). They are based on the assumption that the distribution of the quality characteristic (also called process distribution) is normal or approximately normal. If three sigma control limits are employed for a process that is normal distributed, then the Type I risk of X chart, i.e. the probability of a sample mean X falling outside the control limits when the process is in control will be 0.0027. In fact, many distributions of the industrial processes are asymmetric such as a hot-dip galvanizing, and the semiconductor processes etc. In these cases, the Shewhart control chart will often result in a high false alarm rate.
Some different approaches have been suggested to deal with nonnormal underlying distributions. One approach is to increase the sample size such that the sample mean becomes approximately normally distributed. However, large sample size may not be op-erationally feasible and may increase costs. Another approach is to assume the underlying distribution and the process parameters to be known, and thus the control charts can be constructed by using certain exact methods. In this case, the exact probability limits for the control charts will be derived analytically or approximately for a desired Type I risk. Theoretically, better result may be obtained, but it is rarely used because this approach may be complicated and is difficult to find a distribution to fit the data well.
The third approach is to transform the original data such that the transformed data are more closely modeled by normal distribution, and then the Shewhart control charts are established by using the transformed data. Again, it has not been often used because this method requires knowledge of the underlying distribution and it is usually difficult
to identify an appropriate transformation.
Furthermore, some literature investigated robust control limits based on heuristic methods with no assumptions on the form of the process distributions. Bai and Choi (1995) and Chang and Bai (2001) developed the weighted variance (WV) method and weighted standard deviation (WSD) method, respectively, to set up X and R charts for skewed underlying distributions. Both methods are based on the semivariance approxi-mation of Choobineh and Branting (1986). The idea is that a skewed distribution can be split into two segments at its mean and each segment is used for creating a new symmetric distribution. The two new symmetric distributions created from the original skewed dis-tribution have the same mean but different variances. One of the two created symmetric distributions is used to derive the upper control limit, and the other is used to derive the lower control limit. The WV method decomposes the process variance into two parts, while the WSD method decomposes the process standard deviation into two parts.
Chan and Cui (2003) proposed skewness correction (SC) method to construct X and R control charts by taking into consideration the degree of skewness of the process dis-tribution. The SC method constructs asymmetric control limits by using ±3 standard deviations plus the known function of skewness. It is based on the Cornish-Fisher ex-pansion proposed by Cornish and Fisher (1937). For 0 < α < 1, the Cornish-Fisher expansion shows that the α-quantiles of any distribution F can be expressed in term of the cumulants of F and the α-quantiles of the normal distribution. More precisely, let X be a standardized random variable with mean 0 and standard deviation 1, xα and zα be the upper α-quantiles of the random variable X and the standard normal random variable, respectively. Also let kr be the rth cumulant of X (r ≥ 3). Note that the third cumulant k3 and fourth cumulant k4 are, respectively, the skewness and kurtosis of X.
Then xα has the following Cornish-Fisher expansion : xα = zα+ 1
6(zα2 − 1)k3+ 1
24(z3α− 3zα)k4− 1
36(2zα3 − 5zα)k23· · · .
Throughout this paper, let α1 = 0.0027/2 = 0.00135 and α2 = 1 − 0.0027/2 = 0.99865. It is known that zα1 = 3, zα2 = −3, and the Cornish-Fisher expansions of xα1 and xα2 are
xα1 = 3 + 4 3k3+3
4k4+ · · · ,
and
xα2 = −3 + 4 3k3−3
4k4+ · · · .
Chan and Cui (2003) used ±3+43k3/(1+δk23) to approximate xα1 and xα2, respectively, where δ is a constant. From the simulation results for many skewed distributions, such as Weibull, lognormal and Burr distributions, it shows that δ is closed to 0.2, that is
xα1 ≈ 3 +
4 3k3 1 + 0.2k32,
xα2 ≈ −3 +
4 3k3 1 + 0.2k32.
Hence the asymmetric control limits can be constructed such that the Type I error is closer to 0.0027 of the normal case. Analogous to Chan and Cui (2003), Tadikamalla and Popescu (2007) developed kurtosis correction (KC) method to set up control charts for the symmetric and leptokurtic distributions. For a symmetric random variable X, based on the Cornish-Fisher expansion, Tadikamalla and Popescu (2007) approximated the values of xα1 and xα2 by
xα1 ≈ 3 + k4 1 + 0.33k4
, and
xα2 ≈ −3 − k4
1 + 0.33k4,
respectively, and the control chart can be set up by shifting the traditional Shewhart control limits to both sides by the same amount which is a function of kurtosis.
As the skew normal distributions represent a board class of distributions by its shape parameter, and are flexible in describing the distribution behaviors of quality character-istics, Tsai (2007) proposed the skew normal (SN) method to construct X control limits based on skew normal underlying distributions. It is noted that the skewness k3 of a skew normal distribution lies between −0.9953 and 0.9953.
Recently, inspired by Chan and Cui (2003) and Tadikamalla and Popescu (2007), Wang (2009) proposed skewness and kurtosis correlation (SKC) method to derive control charts with no assumptions on the form of the process distribution. Based on Chan and Cui (2003) and the Cornish-Fisher expansion, Wang (2009) approximated xα1 and xα2 by
xα1 ≈ 3 +
4 3k3 1 + 0.2k32 +
3 4k4 1 + δ|k4|, and
xα2 ≈ −3 +
4 3k3 1 + 0.2k32 −
3 4k4 1 + δ|k4|,
respectively, where δ is a constant. Again using some representative distributions and minimizing the absolute discrepancy between the model and the true quantiles, it is shown that δ is close to 3, and the SKC X and R control charts which take account of skewness and kurtosis at the same time can be constructed.
In this chapter, we will propose a modified skewness and kurtosis correlation (MSKC) method to construct control charts by adjusting the functions of skewness and kurtosis in SKC control charts. In Section 1.2, the MSKC method will be introduced and the MSKC X and R control charts will be developed. In Section 1.3, the performances of the proposed MSKC method are compared with those of SC, WV, Shewhart, WSD and SN methods. Finally, the discussions and conclusions are given in Section 1.4.