Control Charts for the Shape Parameter of Skewed Distribution

The weighted distributions are useful when the sampling is done using an unequal probability of the sampling units. The Weighted Power function distribution (WPFD) has applications in the fields of reliability engineering, management sciences and survival analysis. WPFD is more beneficial in Statistical process control (SPC). SPC is defined as the use of statistical techniques to control a process or production method. SPC tools and procedures can help to monitor process behaviour, discover problems in internal systems, and find solutions for production issues. To identify and remove the variation in different reliability processes and also to monitor the reliability of machines where the number of errors follows WPFD, we develop control charts to keep the process in control. A memory-based control chart like an exponentially weighted moving average (EWMA) control chart and an extended exponentially weighted moving average (EEWMA) control chart are discussed and compared each other. The proposal of these control charts is based on the modified maximum likelihood estimator (MMLE) under the shape parameter of WPFD. We have presented Monte Carlo simulation technique and a real-life application to compare the proposed control charts. This study shows that an EEWMA control chart based on MMLE performs better than EWMA control chart, when the underlying distribution of the errors in process monitoring follows WPFD. These findings can be useful for researchers and practitioners in dealing with production errors and optimizing the output.


Introduction
The different production processes in industries face commonly two types of variations: common cause variation and special cause variation. Common cause variation always exists even if the process is designed very well and maintained very carefully. This variation should be relatively small in magnitude and is uncontrollable and due to many small unavoidable causes. A process is said to be in statistical control if only common cause variation is present. The variations outside this common cause pattern are called In real life scenario, this is not always possible to fulfil the normality assumption for the distribution of error during the process. A very few works in literature is about this situation of not normal process distribution including Noorossana et al. [13], Lin et al. [14], Erto et al. [15], Liang et al. [16] and Ahmed et al. [17]. The EWMA statistic is widely used for shift detection in the ongoing process either to monitor qualitative or quantitative physical phenomena. The generalized form of the existing EWMA statistic was introduced by Naveed et al. [18] and named it as EEWMA.

The Conventional EWMA Control Chart
Let the distribution of the underlying process having the sequence {X t } is normal. Also, let the 0 1, is a known constant. Now EWMA statistics is given by The smoothing constant plays a very important here. As it approaches to zero, it becomes sensitive for a small and moderate shift in mean and as it becomes close to one. It approaches to Shewart control chart. The control limits for EWMA are given below The Traditional Extended Exponentially Weighted Moving Averages Control chart When the distribution of the process is normal, the EEWMA control chart was introduced by Naveed et al. [18]. The EEWMA control chart by Naveed et al. [18] is given as where 0 1 1 and 0 2 1 . T tÀ1 is represents the previous value of the variable and Z tÀ1 denotes the previous value of a statistic.
The mean and variance are given as

Proposed Extended Exponentially Weighted Moving Averages Control Chart Under Non-Normality
Following Zaka et al. [1], it is assumed that the process random variable x 1 ; x 2 ; x 3 ; . . . ; x t are independently and identically distributed following WPFD with probability density function (pdf) and cumulative density function (cdf) for the WPFD are given respectively by , where "β" and "γ" are the scale and shape parameters.
Modified Maximum Likelihood Estimator (MMLE) is used to construct memory less and memorybased control charts to monitor the shape parameter of a process that follows a WPFD. The estimator for the shape parameter of WPFD defined by Zaka et al. [1] is given as, The variance of the b c MMLE is given by Now using Zaka et al. [1] and Naveed et al. [18], the EEWMA statistic is given as For t = 1 For t = 2 By taking expectation, we get By using geometric series, we get So, Var EEW t ð Þ¼m Var EEW t ð Þ¼m The control limits are given as

Algorithmic
Steps a) Generate a random sample of size n = 150 on "X t " from the WPFD are x ¼ bR  (Computed in step (d)). f) Compute ARL value for each EEWMA control chart that based onĉ MMLEðtÞ given that process is incontrol state. g) Now fix ARL 0 = 500 for in-control state of the process and search the suitable value of L, so that ARL 0 for in-control state of process is achieved.
h) Now assume if the process parameter "c" is shifted by from its true value and compute ARL 1 . This step is repeated for different shift values 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.13 and 0.88. Also compute ARL 1 in each case of shift values. i) Plot ARL s values against the values of shift that used in step (g) and (h). j) It is to note that the procedure of EWMA control chart based onĉ MMLEðtÞ observe whether the process following the WPFD is in-control or out of control. If the process is in-control, go to Step (a). Otherwise, record the Run Length, i.e. the process remained in control before it is declared to be out-of-control. k) Repeat this process 5000 times to obtain the ARLs, SDRLs and percentiles.

Results and Discussions
We consider the average run length to compare the performance of the proposed estimators. It is taken as the average number of samples that are used before as the process is out of control. ARL 0 indicates the in control ARL, while the ARL 1 describes the out of control ARL. A chart having a larger ARL 0 and smaller ARL 1 is considered better to be used for the process monitoring.
The Tabs. 1-3 is constructed for ARL values when the parameters of underlying process following WPFD using EWMA and EEWMA. Figs. 1-3 presents the ARL values for EWMA and EEWMA. We observed that EEWMA control chart detects earlier on small shifts as compare to EWMA control chart for different choices of 1 and 2 . The same behaviour is observed from Figs. 1-3.    Figure 1: ARL for the shape parameter of WPFD using EWMA and EEWMA Figure 2: ARL for the shape parameter of WPFD using EWMA and EEWMA Figure 3: ARL for the shape parameter of WPFD using EWMA and EEWMA

Simulation Study
In order to see the working procedure of the proposed control charts, a simulation study was carried out. For this purpose, we generated 25 observations from a WPFD for in-control process, and the next  In Fig. 5, we noted that the proposed EEWMA control chart under MMLE detected a shift at the 32 th sample, while in Fig. 4; the EWMA control chart under MMLE could not detect the shift. Hence, this shows that the proposed EEWMA control chart under MMLE has a greater ability to detect smaller shifts earlier, as compared to the EWMA control chart.

Conclusions
We have discussed the process monitoring for WPFD. In real life, we may face the situation that any specific process does not follow the normal distribution. But the distribution of the errors in the process becomes WPFD. In the current work, we first estimate the parameters of the distribution of the errors during any process by using MMLE, which was claimed better to estimate the parameters of WPFD. By using the MMLE, we then modified the quality control charts used in literature, such as EWMA and EEWMA control charts. We see that EEWMA control chart under MMLE can be used to monitor the process when the underlying distribution of the errors in process monitoring follows WPFD. It is therefore hoped that the findings of this study will be useful for researchers in different fields of applied sciences. It will be helpful to identify the error in time and making strategies to deal with it. Companies can control their cost and improve product quality by applying the proposed quality chart processes. Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.