A study of process monitoring based on inverse Gaussian distribution
Introduction
The control charts are used to monitor and detect the shifts in the process mean and variance of a quality characteristic of interest in a process. The usual Shewhart and R control charts are based on the assumption that the distribution of the observed data is normal. However, there are many cases in which the normality assumption is not valid (Chan and Cui, 2003, Haridy and Ei-Shabrawy, 1996, Roes and Does, 1995). When the underlying distribution is skewed, there are potential problems, namely, the false alarm rates and detection power of an out-of-control condition often substantially differ from what we expect under the normal case (Mahoney, 1998, Tadikamalla et al., 2008). One method used to compute control limits for skewed distributions is to first transform the data to make it quasi-normal and then use the traditional Shewhart control charts (Kao, 2010, Liu et al., 2007, Tadikamalla and Popescu, 2007, Xie et al., 2000). But it is not frequently used because it is difficult to explain the alarm intuitively. Moreover, when the form of the process distribution is known, one may use the exact method, not an approximate method, because the accurate control limits are more likely to detect whether a process is in control (Chan and Cui, 2003, Tadikamalla and Popescu, 2007). Meanwhile, numerous papers show that the performances of the control charts with asymmetric control limits are better than those with symmetric control limits when the underlying distribution is heavily skewed. For some other related discussion, see, e.g., Tagaras, 1989, Quesenberry, 1991, Quesenberry, 1995, Woodall, 1997, Yazici and Kan, 2009, Chen and Kuo, 2010.
The inverse Gaussian distribution is one of the most important skewed distributions because it is highly flexible and there are a few major advantages relative to the other positive skewed distributions (Chhikara and Folks, 1989, Hawkins and Olwell, 1997, Tian and Wilding, 2005, Wu and Li, 2012). Hawkins and Olwell (1997) further gave some motivations to design the control charts to monitor the changes in the parameters of the inverse Gaussian distribution. In the previous studies of the control charts for the inverse Gaussian distribution, Edgeman, 1989a, Edgeman, 1989b constructed the Shewhart-type control charts with the probability limits for detecting the changes in the mean and variability of the inverse Gaussian distribution. Aminzadeh (1993) proposed the control charts with the probability limits for monitoring the mean and dispersion of the inverse Gaussian distribution based on the approximate distribution of the monitoring statistics using the exponential smoothing technique. Based on the sequential probability ratio tests and cumulative sum plans, Edgeman (1996) considered the control chart to monitor the shifts in the location parameter assuming that the shape parameter of the inverse Gaussian distribution is known. Shankar and Joseph (1996) proposed a cumulative sum chart to monitor the mean of the inverse Gaussian distribution assuming that the shape parameter is known. Hawkins and Olwell (1997) developed a CUSUM control chart for the location parameter with the assumption of the fixed shape parameter, and proposed another CUSUM control chart for the shape parameter assuming that the location parameter is fixed. Sim (2003) developed the control chart with the probability limits for the variability under the assumption that the shape parameter is known, namely, the control chart is for the location parameter. Lio and Park (2010) proposed the parametric bootstrap control charts for monitoring the percentiles of the inverse Gaussian distribution.
For the skewed distributions, the mean and variance may be ineffective summary statistics of the process measure (Hawkins & Olwell, 1997), thus we propose the control chart for the shape parameter and the control chart for the location parameter when the shape parameter is in control. Moreover, the Shewhart-type control charts mentioned above are average run length (ARL) biased (For a control chart, if the ARLs under the out-of-control processes are uniformly smaller than that under the in-control process, it is called an ARL-unbiased control chart), which is common for data that follow the skewed distributions (Cheng and Chen, 2011, Guo and Wang, 2014, Zhang et al., 2006). The ARL-biased problem is highly undesirable in practice, since it takes a longer time on average to signal the assignable causes than that when there is no assignable cause. This paper thus focuses on the design of the ARL-unbiased control charts.
This paper is organized as follows. In Section 2, we propose two statistics to monitor the shape and location parameters, respectively. In Section 3, we develop the procedures to design the ARL-unbiased control charts for the shape and location parameters, respectively. In Section 4, we study the effects of parameter estimation on those control charts given in Section 3. In Section 5, we propose a procedure to design the ARL-unbiased control chart with the desired for the shape parameter when the in-control shape parameter is estimated, and study the performance of the proposed control chart. Finally, we use an example to illustrate the proposed control charts.
Section snippets
Monitoring statistics
The probability density function (p.d.f.) of the inverse Gaussian distribution is given bywhere is the shape parameter and is the location parameter. Its mean and variance are and , respectively, therefore the coefficient of variation is . The density is unimodal with shape depending only on (Edgeman, 1996).
Suppose that is a random sample with size n from the distribution.
Let
Design of the control charts with the known parameters
In this section, we shall study how to design the ARL-unbiased control charts for and based on the monitoring statistics and when the in-control values and are known.
The effect of the parameter estimation
The discussion above assumes that the in-control parameters and are known. In most applications, however, the in-control process parameters and are usually unknown and have to be estimated by an in-control Phase I data set. It is well known that the performance of the control charts when the in-control parameters are estimated is different from the known parameters case because of the variability of the estimated parameters. Thus it is essential to study the effects of the parameter
The design of the control chart for the shape parameter when the parameter is estimated
The results of Section 4 show that only when the number m of Phase I samples is large, the performance of the control charts with the estimated parameters is similar to that of the corresponding control charts with the known parameters. In most applications, however, it may not be practical to wait for the accumulation of such large Phase I data set because of the cost or time consideration (Maravelakis and Castagliola, 2009, Teoh et al., 2014, Zhang et al., 2011, Zou et al., 2007). In
An illustrative example
In this section, the example given by Aminzadeh (1993) is used to illustrate the proposed control charts. The data set is shown in Table 11. The data consist of 15 samples of size from . We want to monitor the process changes by means of the proposed ARL-unbiased control charts with the known parameters. Table 11 also gives the values of the monitoring statistics and .
For the known parameters and , we have from the Eqs. (3), (9)
Conclusion
In this paper, we developed the ARL-unbiased control charts to monitor the shifts in the shape and location parameters of the inverse Gaussian distribution when the in-control parameters are known. The proposed control charts are easy to implement and interpret for practitioners.
We also studied the effects of parameter estimation on the proposed control charts. For the shape parameter , we found that when the number of Phase I samples, , the performance of the control chart with the
Acknowledgements
The authors would like to thank the Editor, the Associate Editor and the referees for their detailed comments and suggestions, which helped improve the manuscript. The work of Guo is sponsored by National Social Science Foundation of China (Grant No. 14BTJ030). The work of Wang is supported by Natural Science Foundation of China under the contract number 11371322. Both Guo and Wang are also supported by Zhejiang Provincial Key Research Base for Humanities and Social Science Research (Statistics)
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