Asymmetric control limits for range chart with simple robust estimator under the nonnormal distributed process
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Abstract
This paper aims to modify Shewhart, the weighted variance and skewness correction methods in industrial statistical process control. The robust and asymmetric control limits of range chart are constructed to use in contaminated and skewed distributed process. The way of construction of control limits is simple and corresponds to three methods in which sample range estimator is replaced with the robust interquartile range. These three modified methods are evaluated in terms of their type I risks and average run length by using simulation study. The performance of the proposed range charts is assessed when the Phases I and II data are uncontaminated and contaminated. The Weibull, gamma and lognormal distributions are chosen since they can represent a wide variety of shapes from nearly symmetric to highly skewed.
Keywords
Skewed distributions Shewhart method Weighted variance method Skewness correction method Robust estimatorIntroduction
When the quality variable has a skewed distribution, it might be misleading to observe the process by using the Shewhart \(\bar{X}\) and R control charts. The usage of Shewhart control charts in skewed distributions causes an increase in type I risk (p) when the skewness increases because of the variability in population. For this reason, three methods which use the asymmetric control limits were considered as an alternative to the classical method [13]. The first one is the weighted variance (WV) method proposed by Choobineh and Ballard [6], which is based on the semivariance approximation of Choobineh and Branting [5]. They obtained the asymmetric control limits of \(\bar{X}\) and R charts for skewed distributions based on the standard deviation of sample means and ranges. Bai and Choi [2] also proposed a simple heuristic method of constructing \(\bar{X}\) and R charts by using the WV method. The second one is the weighted standard deviations (WSD) proposed by Chang and Bai [4] to obtain control limits by decomposing the standard deviation into two parts. The last one is a skewness correction (SC) method proposed by Chan and Cui [3] for constructing \(\bar{X}\) and R chart by taking into consideration the degree of skewness of the process distribution, with no assumptions on the distribution. Karagöz and Hamurkaroğlu [13] worked on \(\bar{X}\) and R control charts for skewed distributions which are Weibull, gamma and lognormal. Classical methods of estimating parameters of the distribution of quality characteristic may be affected by the presence of outliers. In order to overcome such situation, robust estimators, which are less affected by the extreme values or small departures from the model assumptions, are introduced in industrial application. AbuShawiesh [1] presented a simple approach to robust estimation of the process standard deviation based on a very robust scale estimator, namely, the median absolute deviation (MAD) from the sample median. The proposed method provides an alternative to the Shewhart S control chart. Schoonhove et al. [19] studied design schemes for the standard deviation control charts with estimated parameters. Different estimators of the standard deviation were considered, and the effect of the estimator on the performance of the control charts under nonnormality was investigated.
Jensen et al. [12] conducted a literature survey of the effects of parameter estimation on control chart properties and identified several issues for future research. The effect of using robust or other alternative estimators has not been studied thoroughly. Most evaluations of performance have considered standard estimators based on the sample mean and standard deviation and have used the same estimators for both Phases I and II. However, in Phase I applications, it seems more appropriate to use an estimator that will be robust to outliers, step changes and other data anomalies. Examples of paper discussing robust estimation methods in Phase I control charts include [7, 16, 17, 25, 26]. One of Jensen et al. [12] their recommendations is to consider the effect of using these robust estimators on Phase II performance. By considering this recommendation, Schoonhove et al. [19] study the impact of these estimators on the Phase II performance of standard deviation control chart.
Recently works on control charts: Sukparungsee [23] studied the robustness of the asymmetric Tukey’s control chart for skew and nonskew distributions as Lognormal and Laplace distributions. The results found that the asymmetric performs better than symmetric Tukey’s control chart for both cases of skew and nonskew process observation. Sindhumol et al. [22] introduced a modification to trimmed standard deviation to increase its efficiency and it is used in controlling process dispersion. Authors constructed a Phase I control chart derived from standard deviation of trimmed mean, which is robust. WeiHeng et al. [11] proposed a new control chart for monitoring the standard deviation of a lognormal process based on the methodologies studied in Tang and Yeh [24]. The fundamental assumption in deriving the approximate confidence intervals in Tang and Yeh [24] was that the variance of the logtransformed normal distribution is less than 1. If the variance is larger than 1, they further derived an approximate confidence interval and develop the control chart accordingly. The proposed chart was compared to the existing charts based on the average run length (ARL), where the run length is defined as the number of samples taken before the first outofcontrol signal shows up on a control chart. Duclos and Pillet [8] proposed the use of a control chart (L chart) build with a minimum variance estimator whose performances have been compared to those of the average in term of variance and distribution shape. They studied this estimator in the case of data incoming from a Multi generator process. Koyuncu and Karagöz [14] proposed to construct the mean control chart limits based on Shewhart, weighted variance and skewness correction methods using simple random sampling, ranked set sampling, median ranked set sampling and neoteric ranked set sampling designs. The performance of the proposed control charts based on neoteric ranked set sampling designs is compared with their counterparts in ranked set sampling, median ranked set sampling and simple random sampling by Monte Carlo simulation.
In this paper, we consider this recommendation to construct asymmetric control limits of R charts under nonnormality and contamination. We propose to modify the Shewhart, WV and SC methods by using the interquartile range estimator of the standard deviation. And we called them modified Shewhart (MS), modified weighted variance (MWV) and modified skewness correction (MSC) methods, respectively. We study on the effect of the robust estimator on control chart performance under nonnormality for moderate sample size (30 subgroups of 5–10). The considered standard estimator is interquartile range. The performance of the estimator is evaluated by assessing their root mean squared error (RMSE) under skewed distribution and in the presence of several types of contamination. Moreover, we derive factors of range control chart for each modified method. The modified robust methods are evaluated in terms of their type I risks and average run length and then compared with the modified Shewhart method. By using Monte Carlo simulation, the p and ARL values of proposed R control charts are compared based on classic and robust estimators. The performance of the proposed robust range charts is assessed when the Phases I and II data are uncontaminated and contaminated skewed distributed process. The Weibull, gamma and lognormal distributions are chosen since they can represent a wide variety of shapes from nearly symmetric to highly skewed. Khodabin and Ahmadabadi [10] was introduced the generalized gamma (GG) distribution that is a flexible distribution in statistical literature, and has exponential, gamma, and Weibull as subfamilies, and lognormal as a limiting distribution.
The remainder of the paper is structured as follows. The next section presents the design schemes and gives the methods. In the subsequent “Measuring estimator’s efficiency” section, the efficiency of measuring estimators is described and the control chart constants are given in “Determination of control charts constants” section. The performance of methods is evaluated in “The performance of the modified methods” section by considering simulation study. “Results” section evaluates the results of the study. Finally, a conclusion of this study is given in “Conclusion” section.
Skewed distributions, estimators and modified methods
The main interest of this section is to give all mathematical details by regarding the robust R control charts for skewed distributions. Firstly, the skewed distributions are discussed in “Skewed distributions” section. Secondly, the classic and robust estimators are given in “Classic and robust estimators” section. We propose to modify Shewhart, WV and SC methods by replacing the mean of the subgroup ranges with the mean of the subgroup interquartile ranges. And finally, the modified methods based on robust estimator for skewed distributions are given in “Modified methods” section.
Skewed distributions
 The probability density function of the Weibull distribution is defined asfor \(x>0\), where \(\beta\) is a shape parameter and \(\lambda\) is a scale parameter.$$\begin{aligned} f(x\beta ,\lambda )=\beta \lambda ^\beta {x}^{\beta 1}\exp (x\lambda )^\beta \end{aligned}$$
 The probability density function of the gamma distribution is defined asfor \(x>0\), where \(\alpha\) is a shape parameter and \(\beta\) is a scale parameter.$$\begin{aligned} f(x\alpha ,\beta )=\frac{1}{\Gamma (\alpha )\beta ^\alpha }x^{\alpha 1}\exp \left( \frac{x}{\beta }\right) \end{aligned}$$
 The probability density function of the lognormal distribution is defined asfor \(x>0\), where \(\sigma\) is a scale parameter and \(\mu\) is a location parameter.$$\begin{aligned} f(x\sigma ,\mu )=\frac{1}{x\sigma \sqrt{2\pi }}\exp \left( \frac{({{\text{ln}}(x)\mu })^2}{2\sigma ^2}\right) \end{aligned}$$
Classic and robust estimators
The process is assumed to be in control (i.e., in Phase I) with given \(\hat{\sigma }\) . The process parameters \(\mu\) and \(\sigma\) are estimated from samples, and the resulting estimates are used to monitor the process in Phase II. We define \(\hat{\mu }\) and \(\hat{\sigma }\) as unbiased estimates of \(\mu\) and \(\sigma\), respectively, based on the number of sample k.
Modified methods
In this section, we construct the control limits of R control chart by considering modification in the Shewhart, WV and SC methods. The control limits are derived by assuming that the parameters of the process are unknown. What actually we do is to use simple robust estimator in these three models under the contaminated skewed process. These proposed models are called the MS, MWV and MS methods. When the control limits of MS are symmetric for normal distributed process, the control limits of MWV and MSC are asymmetric for the skewed distributed process.
The MS method
The MWV method
The WV method was proposed by Choobineh and Ballard [6]. The WV method decomposes the skewed distribution into two parts at its mean, and both parts are considered symmetric distributions which have the same mean and different standard deviation. In this method, \(\mu _{R}\) is normally estimated using the mean of the subgroup ranges \(\bar{R}\).
The MSC method
The last method being considered is the SC method proposed by Chan and Cui [3]. They proposed to construct the \(\bar{X}\) and R control charts limits for SC method under the skewed distributions. It’s asymmetric control limits are obtained by taking into consideration the degree of skewness estimated from subgroups and making no assumptions about distributions.
Simulation study
The considered standard deviation estimator is interquartile range. The performance of the estimator is evaluated by assessing their RMSE under skewed distribution and in the presence of several types of contamination. The simulation studies evaluate the efficiency of measuring estimator in “Measuring estimator’s efficiency” section, the control chart constants in “Determination of control charts constants” section and the performance of modified methods in “The performance of the modified methods” section.
Measuring estimator’s efficiency

Model 1: The reference distribution parameters are selected with respect to skewness of distribution that is given in Table 1.

Model 2: The case of 10% replacement outliers coming from another Weibull distribution with a different scale parameter (\(\lambda _1=0.2\)) and a shape parameter \((\beta _1={0.2*\beta })\) , another lognormal distribution with a different location parameter (\(\mu _1=0.2\)) and a scale parameter \((\sigma _1=2*\sigma )\) and another gamma distribution with a different shape parameter (\(\alpha _1=2\alpha\)) and a scale parameter \((\beta _1=0.2)\).

Model 3: The case of 10% replacement outliers from a uniform distribution on [0, 20].

Model 4: The more extreme case of 10% of outliers placed at 50. We replace 10% of observations from the data with extreme values such as 50 to create a outliers in the data.
Values of the skewness and the parameters of distributions
\(k_3\)  0.50  1.00  1.50  2.00  2.50  3.00  

Weibull  \(\beta\)  2.15  1.57  1.20  1.00  0.86  0.77 
Lognormal  \({\sigma }\)  0.16  0.32  0.44  0.54  0.66  0.72 
Gamma  \(\alpha\)  16.00  4.00  1.80  1.00  0.64  0.44 
RMSE of the scale (\(\sigma\)) estimator under the skewed distributions for \(n = 5,10\)
Model/\(k_3\)  \(n=5\)  \(n=10\)  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
Weibull distribution  
Model 1  
Classic  0.0293  0.0446  0.0712  0.1048  0.1520  0.2036  0.1363  0.1899  0.2742  0.3741  0.5095  0.6592 
Robust  0.0337  0.0491  0.0738  0.1042  0.1460  0.1924  0.0290  0.0409  0.0623  0.0934  0.1453  0.2117 
Model 2  
Classic  0.0315  0.0492  0.0799  0.1176  0.1707  0.2291  0.1467  0.2093  0.3067  0.4230  0.5791  0.7509 
Robust  0.0382  0.0578  0.0901  0.1298  0.1836  0.2432  0.0347  0.0501  0.0686  0.0873  0.1154  0.1511 
Model 3  
Classic  1.5846  1.5864  1.5966  1.6086  1.6191  1.6268  2.5950  2.6168  2.6653  2.7317  2.8020  2.8895 
Robust  0.7251  0.7575  0.8178  0.8898  0.9804  1.0721  0.0591  0.0893  0.1281  0.1664  0.2067  0.2429 
Model 4  
Classic  8.4085  8.5658  8.8463  9.1533  9.5031  9.8179  13.3728  13.6361  14.1158  14.6384  15.2350  15.7725 
Robust  3.6826  3.8093  4.0516  4.3354  4.6874  5.0316  0.0621  0.0942  0.1386  0.1826  0.2323  0.2821 
Lognormal distribution  
Model 1  
Classic  0.0314  0.0752  0.1247  0.1839  0.2902  0.3610  0.1481  0.3266  0.5037  0.6972  1.0141  1.2212 
Robust  0.0352  0.0798  0.1263  0.1777  0.2675  0.3269  0.0298  0.0685  0.1118  0.1679  0.2716  0.3461 
Model 2  
Classic  0.1640  0.1279  0.1631  0.2442  0.4417  0.6190  0.3744  0.4560  0.6217  0.8501  1.3030  1.6448 
Robust  0.1018  0.1089  0.1473  0.2064  0.3297  0.4311  0.0394  0.0669  0.1034  0.1528  0.2474  0.3152 
Model 3  
Classic  1.3094  1.2203  1.1497  1.0904  1.0209  0.9931  2.1632  2.1307  2.1254  2.1454  2.2356  2.3170 
Robust  0.6167  0.6405  0.6767  0.7213  0.7938  0.8367  0.0519  0.1131  0.1655  0.2109  0.2725  0.3057 
Model 4  
Classic  8.0412  8.0930  8.1701  8.2778  8.4516  8.5472  12.8626  13.0311  13.2564  13.5063  13.9106  14.1228 
Robust  3.5883  3.7202  3.8852  4.0817  4.4061  4.6041  0.0557  0.1308  0.2028  0.2763  0.3863  0.4468 
Gamma distribution  
Model 1  
Classic  0.2813  0.1570  0.1203  0.1048  0.0966  0.0927  1.3256  0.6772  0.4724  0.3748  0.3227  0.2938 
Robust  0.3183  0.1698  0.1241  0.1045  0.0946  0.0896  0.2693  0.1425  0.1062  0.0940  0.0941  0.1023 
Model 2  
Classic  0.8501  0.1626  0.1228  0.1089  0.1004  0.0953  2.3528  0.7093  0.4487  0.3469  0.2969  0.2681 
Robust  0.6610  0.1803  0.1261  0.1098  0.0998  0.0939  0.3639  0.1428  0.1130  0.1143  0.1192  0.1267 
Model 3  
Classic  0.5513  0.8855  1.3340  1.6147  1.8183  2.0093  1.9356  1.8544  2.3800  2.7347  3.0083  3.2867 
Robust  0.4404  0.5739  0.7623  0.8931  1.0037  1.1248  0.2813  0.2239  0.1900  0.1663  0.1454  0.1270 
Model 4  
Classic  4.9489  7.6274  8.4630  9.1127  9.7395  10.4095  8.7237  12.5101  13.6815  14.6295  15.5769  16.6415 
Robust  2.5862  3.6174  3.9768  4.3167  4.7097  5.1889  0.5141  0.2969  0.2233  0.1815  0.1535  0.1318 
 (i)
When there is no contamination for small sample size, the efficiency of the classic and robust estimators is more or less similar. However, for the large sample size, the robust estimator of scale performs better than the classic estimator when no contamination is present.
 (ii)
Contamination by extreme outliers causes a large increase in the RMSE of the classical estimator, especially for large samples \(n=10\) and a much smaller increase in the RMSE of the robust alternative. The fact that the best performing estimator is robust one, when diffuse outlier disturbances is present for large sample sizes.
 (iii)
For the scale estimation, the interquartile range estimator performs for large sample size better than the small sample size, especially in contamination by extreme outliers for all considered distributions.
 (iv)
In the presence of outliers, the classic scale estimator has the highest RMSE of all skewed distributions.
 (v)
For three skewed distributions, the robust scale estimator has a lower RMSE than the classical in all contaminated cases considered. So it is seen that the robust estimator is more efficient than the classic estimator.
Determination of control charts constants
The constants \(d_2, d_3\) and \(d_4\) are considered under nonnormality to correct the control chart limits. The corrected constants are determined such that the expected value of the statistic divided by the constant is equal to the true value of \(\sigma\). The WV method constants \(d_{2}^{*}\) and \(d_{3}^{*}\) were calculated by taking the mean and standard deviation of range \(\left( \frac{R}{\sigma }\right)\), respectively. In this study, we consider the MS and MWV methods constants \(d_{2}^{Q }\) and \(d_{3}^{Q }\) which are calculated by taking the mean and standard deviation of interquartile range \(\left( \frac{{\text {IQR}}}{\sigma }\right)\), respectively. The SC method constant \(d_{4}^{*}\) is calculated by using Eq. (2.18). We consider the MSC method constant \(d_{4}^{Q }\) which is calculated by using Eq. (2.21).
Values of constants for the skewed distributions
\(k_3\)  Weibull  Lognormal  Gamma  

\(d_2^{*}\)  \(d_3^{*}\)  \(d_4^{*}\)  \(d_2^{*}\)  \(d_3^{*}\)  \(d_4^{*}\)  \(d_2^{*}\)  \(d_3^{*}\)  \(d_4^{*}\)  
\(n=5\)  
0.50  2.3088  0.8493  0.5553  2.3092  0.8948  0.6919  2.3089  0.8889  0.6738 
1.00  2.2559  0.9377  0.8193  2.2575  0.9843  0.9825  2.2595  0.9629  0.9108 
1.50  2.1702  1.0690  1.0564  2.1974  1.0771  1.1630  2.1827  1.0661  1.0873 
2.00  2.0831  1.1859  1.1998  2.1346  1.1644  1.2666  2.0827  1.1852  1.1991 
2.50  1.9903  1.2950  1.2955  2.0423  1.2765  1.3456  1.9758  1.3023  1.2729 
3.00  1.9102  1.3822  1.3501  1.9911  1.3315  1.3675  1.8621  1.4120  1.3275 
\(n=10\)  
0.50  3.0213  0.7667  0.5034  3.0640  0.8442  0.6495  3.0587  0.8335  0.6241 
1.00  2.9709  0.8902  0.7733  3.0225  0.9786  0.9696  3.0050  0.9374  0.8780 
1.50  2.8990  1.0706  0.9946  2.9701  1.1162  1.1463  2.9258  1.0759  1.0360 
2.00  2.8301  1.2342  1.1294  2.9145  1.2490  1.2502  2.8287  1.2337  1.1303 
2.50  2.7530  1.3933  1.2305  2.8300  1.4182  1.3278  2.7323  1.3870  1.1928 
3.00  2.6842  1.5247  1.2943  2.7806  1.5061  1.3525  2.6348  1.5346  1.2445 
Values of robust constants for the skewed distributions
\(k_3\)  Weibull  Lognormal  Gamma  

\(d_2^Q\)  \(d_3^Q\)  \(d_4^Q\)  \(d_2^Q\)  \(d_3^Q\)  \(d_4^Q\)  \(d_2^Q\)  \(d_3^Q\)  \(d_4^Q\)  
\(n =5\)  
0.50  1.3332  0.5686  0.7739  1.3094  0.5682  0.8198  1.3122  0.5684  0.8144 
1.00  1.2921  0.5918  0.8998  1.2665  0.5860  0.9534  1.2773  0.5880  0.9326 
1.50  1.2201  0.6244  1.0481  1.2166  0.6029  1.0670  1.2218  0.6161  1.0529 
2.00  1.1459  0.6504  1.1545  1.1656  0.6177  1.1583  1.1457  0.6502  1.1552 
2.50  1.0672  0.6709  1.2399  1.0932  0.6346  1.2500  1.0587  0.6821  1.2339 
3.00  1.0003  0.6846  1.2970  1.0539  0.6416  1.2868  0.9635  0.7088  1.2968 
\(n=10\)  
0.50  1.3415  0.4825  0.5630  1.2928  0.4733  0.6166  1.2982  0.4747  0.6129 
1.00  1.2892  0.4920  0.6993  1.2350  0.4756  0.7484  1.2572  0.4829  0.7313 
1.50  1.1929  0.5037  0.8752  1.1698  0.4760  0.8701  1.1893  0.4948  0.8742 
2.00  1.0930  0.5096  1.0089  1.1030  0.4738  0.9702  1.0926  0.5092  1.0098 
2.50  0.9875  0.5084  1.1177  1.0113  0.4665  1.0768  0.9786  0.5218  1.1172 
3.00  0.8980  0.5027  1.1923  0.9617  0.4608  1.1237  0.8504  0.5284  1.2127 
The performance of the modified methods
When the parameters of the process are unknown, control charts can be applied in a twophase procedure. In Phase I, control charts are used to define the incontrol state of the process and to assess process stability for ensuring that the reference sample is representative of the process. The parameters of the process are estimated from Phase I sample, and control limits are estimated for using in Phase II. In Phase II, samples from the process are prospectively monitored for departures from the incontrol state. The p indicates the probability of a subgroup range falling outside the control limits. The ARL is the number of points plotted within the control limits before one exceeds the limits. The ARL is the most common measure of control chart performance, and much of it is popularity is due to it is intuitively appealing and more widely applicable.
In this section, we consider design schemes for the R control chart for noncontaminated and contaminated skewed distributed data. We use the mean and the trimmed mean estimators of mean and the range and the interquartile range estimators of the standard deviation for considered methods. To evaluate the control chart performance, we obtain p and ARL for moderate sample size (30 subgroups of 3–10) for each skewed distribution. Control charts can be applied in a twostage procedure, when the parameters of a quality characteristic of the process are unknown. In Phase I, control charts are used to study a historical data set and determine the samples that are out of control. On the basis of the resulting reference sample, the process parameters are estimated and control limits are calculated for Phase II. In Phase II, control charts are used for realtime process monitoring [21].
The simulation consists of two phases is run by using MATLAB R2013. The steps of each phase are described as follows.
 1.a.
Generate n i.i.d. Weibull \((\beta ,1)\), gamma \((\alpha ,1)\) and lognormal \((0,\sigma )\) varieties for \(n=3,5,7,10\).
 1.b.
Repeat step 1.a 30 times \(\left( k=30\right) .\)
 1.c.
By using classic estimators, compute the control limits for Shewhart, the WV and the SC methods. By using robust estimators, compute the control limits for the MS, the MWV and the MSC methods.
 2.a.
Generate n i.i.d. Weibull \((\beta ,1)\), gamma \((\alpha ,1)\) and lognormal \((0,\sigma )\) varieties using the procedure of step 1.a.
 2.b.
Repeat step 2.a 100 times (\(k=100\)).
 2.c.
Compute the sample statistics for R chart for the Shewhart, WV and SC methods. Compute the robust estimator interquartile range IQR for the MS, MWV and MSC methods.
 2.d.
Record whether or not the sample statistics calculated in step 2.c are within the control limits of step 1.c. for all methods.
 2.e.
Repeat steps 1.a through 2.d, 100.000 times and obtain p and ARL values for each method.

Noncontaminated case: The reference distribution parameters are selected with respect to skewness of distribution given in Table 1.

Contaminated case: The more extreme case of 10% of outliers placed at 50. We consider the contamination in Phases I and II.
Results of p and ARL values for the R control chart under the skewed distributions for \(n=5,10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Weibull  
Shewhart  0.0090  0.0110  0.0136  0.0157  0.0177  0.0191  150.8978  98.1740  72.3275  59.6196  53.4702  47.9846 
WV  0.0077  0.0084  0.0096  0.0107  0.0118  0.0125  181.8512  132.5381  102.8912  88.8415  80.8865  72.4061 
SC  0.0023  0.0028  0.0036  0.0045  0.0053  0.0064  427.7160  351.2469  274.0477  223.9642  190.4399  156.0549 
Lognormal  
Shewhart  0.0084  0.0127  0.0157  0.0177  0.0195  0.0204  119.0660  78.9634  63.8602  56.6409  51.2424  49.1268 
WV  0.0073  0.0102  0.0122  0.0134  0.0145  0.0151  136.2565  98.2000  82.1112  74.4801  68.8478  66.4143 
SC  0.0028  0.0041  0.0049  0.0055  0.0065  0.0071  350.9141  243.1315  202.5481  181.6233  153.6098  139.9835 
Gamma  
Shewhart  0.0051  0.0090  0.0143  0.0166  0.0183  0.0181  197.3515  111.4107  70.1326  60.1030  54.5786  55.2337 
WV  0.0042  0.0066  0.0099  0.0112  0.0121  0.0117  240.1825  150.7409  100.6765  89.3152  82.8363  85.5498 
SC  0.0014  0.0023  0.0036  0.0044  0.0053  0.0054  715.5123  429.7194  274.0777  226.0551  189.4191  186.1851 
\(n=10\)  
Weibull  
Shewhart  0.0062  0.0094  0.0124  0.0151  0.0169  0.0183  160.6942  106.6553  80.4959  66.0153  59.0493  54.6209 
WV  0.0052  0.0069  0.0085  0.0099  0.0108  0.0116  192.9012  144.7807  117.8134  101.0816  92.3702  86.3707 
SC  0.0035  0.0039  0.0041  0.0044  0.0048  0.0053  283.9296  259.4707  243.6647  229.2526  208.3333  188.4659 
Lognormal  
Shewhart  0.0080  0.0122  0.0150  0.0169  0.0186  0.0194  125.7182  82.1970  66.7111  59.3246  53.6579  51.5730 
WV  0.0069  0.0097  0.0115  0.0127  0.0137  0.0141  145.4376  102.7485  86.6371  78.9011  72.9442  70.9441 
SC  0.0040  0.0054  0.0058  0.0058  0.0062  0.0066  252.2513  185.4599  171.0864  171.8715  161.6501  150.9799 
Gamma  
Shewhart  0.0040  0.0076  0.0127  0.0148  0.0168  0.0171  252.8957  131.0547  78.9004  67.5466  59.5334  58.5888 
WV  0.0032  0.0055  0.0086  0.0097  0.0107  0.0107  312.2171  180.9431  115.7823  103.4148  93.3315  93.8183 
SC  0.0019  0.0031  0.0045  0.0043  0.0048  0.0051  539.2580  327.1502  223.8489  234.6702  207.3183  196.7420 
Results of p and ARL values for the R control chart under the contaminated skewed distributions for \(n=5,10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Weibull  
Shewhart  0.3638  0.2773  0.1483  0.0537  0.0101  0.0016  0.0003  0.0004  0.0007  0.0019  0.0099  0.0631 
WV  0.7537  0.6602  0.6187  0.6005  0.3984  0.0357  0.0001  0.0002  0.0002  0.0002  0.0003  0.0028 
SC  0.8663  0.6513  0.1025  0.0006  0.0000  0.0000  0.0001  0.0002  0.0010  0.1595  9.9010  9.0909 
Lognormal  
Shewhart  0.3569  0.2587  0.0935  0.0198  0.0014  0.0002  0.0000  0.0000  0.0000  0.0001  0.0007  0.0052 
WV  0.7537  0.6283  0.5957  0.3833  0.0062  0.0001  0.0000  0.0000  0.0000  0.0000  0.0002  0.0179 
SC  0.7775  0.4556  0.1713  0.0175  0.0000  0.0000  0.0000  0.0000  0.0000  0.0001  1.2500  0.3333 
Gamma  
Shewhart  0.2829  0.2531  0.1458  0.0544  0.0126  0.0018  0.0000  0.0000  0.0001  0.0002  0.0008  0.0056 
WV  0.6596  0.6300  0.6142  0.6004  0.5560  0.2455  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000 
SC  0.3019  0.3242  0.1533  0.0007  0.0001  0.0000  0.0000  0.0000  0.0001  0.0152  0.1420  1.4493 
\(n=10\)  
Weibull  
Shewhart  0.4183  0.3774  0.0004  0.0000  0.0000  0  0.0000  0.0000  0.0245  0.7752  1.5873  Inf 
WV  0.3715  0.3687  0.3686  0.3687  0.3668  0.3175  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000 
SC  0.3827  0.3691  0.3686  0.3594  0.2119  0.0051  0.0000  0.0000  0.0000  0.0000  0.0000  0.0019 
Lognormal  
Shewhart  0.4050  0.0033  0.0000  0.0000  0.0000  0.0000  0.0000  0.0030  0.2833  Inf  8.3333  2.0833 
WV  0.3689  0.3686  0.3686  0.3590  0.1266  0.0075  0.0000  0.0000  0.0000  0.0000  0.0001  0.0013 
SC  0.3713  0.3687  0.3658  0.3292  0.1363  0.0268  0.0000  0.0000  0.0000  0.0000  0.0001  0.0004 
Gamma  
Shewhart  0.3292  0.0404  0.0004  0.0000  0.0000  0.0000  0.0003  0.0025  0.2684  5.1546  Inf  Inf 
WV  0.3690  0.3687  0.3687  0.3685  0.3686  0.3686  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003 
SC  0.3687  0.3684  0.3679  0.3592  0.2221  0.0000  0.0003  0.0003  0.0003  0.0003  0.0005  Inf 
Results of p and ARL values for the modified R control chart under the skewed distributions for \(n=5, 10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Weibull  
MS  0.0068  0.0101  0.0140  0.0166  0.0190  0.0207  110.8697  91.0349  73.5456  63.6217  56.3784  52.3714 
MWV  0.0056  0.0074  0.0097  0.0111  0.0125  0.0135  129.1222  118.8241  103.7151  93.4588  84.5559  79.7550 
MSC  0.0023  0.0027  0.0034  0.0040  0.0048  0.0054  441.3647  370.3978  295.8667  249.5882  207.9910  185.1269 
Lognormal  
MS  0.0097  0.0119  0.0140  0.0156  0.0174  0.0183  103.0460  84.1085  71.2550  64.1420  57.3076  54.6397 
MWV  0.0086  0.0096  0.0109  0.0117  0.0128  0.0133  116.2601  104.1873  92.1328  85.2123  77.9460  75.2417 
MSC  0.0025  0.0031  0.0037  0.0043  0.0050  0.0054  402.9496  321.8228  266.9585  231.3958  199.0129  183.6311 
Gamma  
MS  0.0096  0.0115  0.0136  0.0158  0.0176  0.0195  104.0832  86.7965  73.3003  63.3136  56.7032  51.1593 
MWV  0.0085  0.0091  0.0099  0.0108  0.0114  0.0122  118.3124  109.7020  100.6867  92.8902  87.9825  82.0371 
MSC  0.0025  0.0029  0.0035  0.0041  0.0046  0.0055  406.3389  342.9120  289.1093  245.2784  215.5172  183.3954 
\(n=10\)  
Weibull  
MS  0.0065  0.0083  0.0107  0.0129  0.0149  0.0163  154.6671  120.3891  93.1359  77.2630  67.2079  61.2985 
MWV  0.0058  0.0069  0.0084  0.0098  0.0110  0.0119  171.5031  144.8834  118.9061  102.0169  90.9579  84.0555 
MSC  0.0022  0.0026  0.0030  0.0034  0.0039  0.0043  454.5661  387.3717  332.7123  293.6082  257.9580  230.4466 
Lognormal  
MS  0.0072  0.0091  0.0108  0.0122  0.0141  0.0149  139.3942  109.9989  93.0103  81.9618  71.1187  67.1686 
MWV  0.0066  0.0078  0.0090  0.0099  0.0112  0.0118  151.5129  127.4551  111.7069  100.5935  89.1353  84.8097 
MSC  0.0024  0.0029  0.0033  0.0037  0.0041  0.0044  418.7079  342.8415  299.8950  268.6583  243.4808  229.8586 
Gamma  
MS  0.0072  0.0087  0.0108  0.0128  0.0149  0.0170  139.8562  114.4505  92.8402  78.0616  66.9035  58.7492 
MWV  0.0066  0.0074  0.0086  0.0097  0.0108  0.0118  152.3693  134.6928  116.2966  103.2439  92.7756  84.7214 
MSC  0.0024  0.0027  0.0031  0.0033  0.0038  0.0045  418.3750  367.4984  324.8019  299.8591  261.7116  223.7086 
Results of p and ARL values for the R control chart for contaminated Weibull distrubuted data for \(n=5,10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Model 1  
MS  0.0073  0.0085  0.0103  0.0119  0.0135  0.0147  137.6349  118.2033  96.6828  84.2950  74.0351  68.2058 
MWV  0.0063  0.0064  0.0072  0.0078  0.0086  0.0092  159.7393  155.6735  139.6980  127.8462  115.6109  108.1210 
MSC  0.0016  0.0017  0.0021  0.0025  0.0030  0.0035  631.7918  572.4098  473.1040  399.7921  332.4468  288.5087 
Model 2  
MS  0.1577  0.1276  0.0835  0.0494  0.0297  0.0220  6.3413  7.8361  11.9811  20.2308  33.6828  45.4339 
MWV  0.2869  0.1092  0.0454  0.0236  0.0143  0.0111  3.4851  9.1608  22.0294  42.4196  69.9208  89.9612 
MSC  0.0034  0.0045  0.0056  0.0061  0.0068  0.0076  294.1176  222.2222  178.5714  163.9344  147.0588  131.5789 
Model 3*  
MS  0.2591  0.1619  0.0616  0.0217  0.0092  0.0063  0.0039  0.0062  0.0162  0.0461  0.1083  0.1596 
MWV  0.6400  0.6168  0.5944  0.4955  0.2309  0.0315  0.0016  0.0016  0.0017  0.0020  0.0043  0.0317 
MSC  0.0033  0.0041  0.0054  0.0061  0.0071  0.0076  303.0303  243.9024  185.1852  163.9344  140.8451  131.5789 
\(n=10\)  
Model 1  
MS  0.0052  0.0065  0.0084  0.0102  0.0119  0.0132  190.8543  152.9847  119.2521  98.3362  84.3633  75.7714 
MWV  0.0047  0.0054  0.0065  0.0075  0.0085  0.0093  211.7164  185.8322  154.4879  132.9770  117.0412  107.0652 
MSC  0.0016  0.0018  0.0020  0.0022  0.0026  0.0030  607.2014  551.0856  498.0080  446.3489  385.8769  335.7057 
Model 2  
MS  0.0088  0.0117  0.0153  0.0177  0.0198  0.0211  114.1657  85.3446  65.3104  56.6200  50.6201  47.4404 
MWV  0.0076  0.0095  0.0118  0.0133  0.0145  0.0153  132.4468  105.5086  84.5931  75.2950  68.7753  65.1831 
MSC  0.0033  0.0042  0.0050  0.0056  0.0062  0.0066  304.5995  237.2198  198.2043  179.3207  162.3034  151.6139 
Model 3  
MS  0.0084  0.0117  0.0156  0.0185  0.0210  0.0228  118.8736  85.4672  64.1997  53.9476  47.5563  43.8416 
MWV  0.0072  0.0094  0.0119  0.0139  0.0155  0.0166  139.2641  106.7954  83.7318  72.1511  64.6914  60.4186 
MSC  0.0032  0.0042  0.0052  0.0059  0.0068  0.0075  316.3456  239.8772  191.7068  168.1775  146.5159  133.5381 
Results of p and ARL for the R control chart for contaminated lognormal distrubuted data for \(n=5,10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Model 1*  
MS  0.2332  0.0928  0.0355  0.0192  0.0134  0.0126  4.2883  10.7794  28.1769  52.0115  74.4718  79.6292 
MWV  0.6302  0.5951  0.4382  0.1759  0.0108  0.0041  1.5867  1.6803  2.2818  5.6865  92.3898  244.5287 
MSC  0.1151  0.0156  0.0037  0.0017  0.0019  0.0022  8.6847  64.1355  270.2995  588.0969  532.0847  452.7755 
Model 2  
MS  0.2342  0.0918  0.0356  0.0193  0.0135  0.0126  4.2697  10.8977  28.0793  51.6994  74.0516  79.3789 
MWV  0.6307  0.5954  0.4379  0.1757  0.0107  0.0041  1.5856  1.6796  2.2837  5.6921  93.4789  243.3623 
MSC  0.1156  0.0155  0.0037  0.0017  0.0019  0.0022  8.6472  64.3488  266.8446  577.1673  528.3737  452.2840 
Model 3  
MS  0.2339  0.0925  0.0353  0.0192  0.0134  0.0126  4.2753  10.8137  28.3280  52.0869  74.3732  79.4407 
MWV  0.6305  0.5953  0.4379  0.1763  0.0108  0.0041  1.5860  1.6799  2.2835  5.6721  92.9964  242.6007 
MSC  0.1154  0.0156  0.0037  0.0017  0.0019  0.0022  8.6678  64.0541  269.0125  587.8895  526.2327  455.1869 
\(n=10\)  
Model 1  
MS  0.0066  0.0084  0.0104  0.0124  0.0145  0.0157  151.0688  119.0023  95.7781  80.9042  68.8463  63.7357 
MWV  0.0064  0.0074  0.0088  0.0102  0.0117  0.0125  156.2891  135.3034  113.8330  98.4581  85.6362  80.0109 
MSC  0.0021  0.0026  0.0033  0.0038  0.0045  0.0048  467.3334  380.7058  307.4274  264.3614  224.6383  209.0957 
Model 2  
MS  0.0096  0.0133  0.0159  0.0177  0.0193  0.0197  103.7700  75.4484  62.8460  56.5294  51.7582  50.8536 
MWV  0.0085  0.0112  0.0130  0.0142  0.0152  0.0154  117.5834  89.6781  76.6554  70.1887  65.5832  64.9642 
SC  0.0037  0.0050  0.0060  0.0065  0.0066  0.0065  273.8901  199.0723  167.1682  154.9571  150.6546  152.8608 
Model 3  
MS  0.0097  0.0137  0.0170  0.0195  0.0223  0.0237  102.7485  72.8444  58.8433  51.1679  44.7507  42.2060 
MWV  0.0085  0.0114  0.0137  0.0156  0.0175  0.0185  118.2033  87.9245  72.9325  64.2731  57.0145  54.0862 
MSC  0.0037  0.0053  0.0065  0.0074  0.0084  0.0089  272.4796  189.8686  153.4425  134.7600  118.5354  111.9545 
Results of p and ARL for the R control chart for contaminated gamma distrubuted data for \(n=5,10\)
Method/\(k_3\)  p Values  ARL values  

0.5  1.0  1.5  2  2.5  3.0  0.5  1.0  1.5  2  2.5  3.0  
\(n=5\)  
Model 1*  
MS  0.0126  0.0298  0.0337  0.0218  0.0096  0.0032  0.0080  0.0034  0.0030  0.0046  0.0105  0.0317 
MWV  0.0042  0.2180  0.4449  0.4953  0.4653  0.3109  0.0237  0.0005  0.0002  0.0002  0.0002  0.0003 
MSC  0.0023  0.0033  0.0015  0.0005  0.0001  0.0000  0.0434  0.0304  0.0646  0.2147  1.0460  8.4746 
Model 2  
MS  0.0070  0.0108  0.0149  0.0175  0.0199  0.0215  142.3751  92.7154  67.3260  57.0002  50.2535  46.6105 
MWV  0.0073  0.0085  0.0106  0.0118  0.0129  0.0134  136.3066  117.6277  93.9541  84.4466  77.6108  74.6001 
MSC  0.0015  0.0027  0.0039  0.0048  0.0056  0.0063  676.1782  375.6574  257.1818  208.6202  178.3930  159.2433 
Model 3  
MS  0.0099  0.0227  0.0394  0.0492  0.0506  0.0463  100.5257  44.1117  25.3756  20.3405  19.7551  21.5964 
MWV  0.0096  0.0157  0.0223  0.0234  0.0206  0.0160  103.7990  63.6821  44.7950  42.8016  48.5765  62.6586 
MSC  0.0025  0.0069  0.0107  0.0112  0.0097  0.0072  394.7888  144.8792  93.8738  89.2746  103.1481  139.5615 
\(n=10\)  
Model 1  
MS  0.0065  0.0077  0.0101  0.0137  0.0169  0.0188  153.6523  129.8634  99.3631  72.8810  59.2466  53.0729 
MWV  0.0063  0.0067  0.0081  0.0103  0.0122  0.0132  159.9488  149.1892  123.9572  96.7455  82.1450  75.6630 
MSC  0.0021  0.0023  0.0028  0.0037  0.0046  0.0053  475.3078  433.3131  354.9624  269.2877  216.7458  189.5735 
Model 2  
MS  0.0070  0.0119  0.0154  0.0178  0.0194  0.0208  142.6595  84.3398  65.0830  56.2506  51.4279  48.1621 
MWV  0.0066  0.0099  0.0121  0.0134  0.0140  0.0143  150.8978  100.9000  82.3621  74.7138  71.3353  69.8944 
MSC  0.0023  0.0043  0.0053  0.0056  0.0059  0.0062  427.5514  233.7923  189.9407  177.8126  168.2737  160.1640 
Model 3  
MS  0.0096  0.0128  0.0159  0.0185  0.0203  0.0217  103.8217  77.9065  62.7668  54.0728  49.3167  46.0259 
MWV  0.0083  0.0105  0.0124  0.0139  0.0145  0.0151  120.0999  95.4399  80.4117  72.1975  68.9280  66.3883 
MSC  0.0036  0.0048  0.0056  0.0060  0.0063  0.0067  276.7323  210.1370  180.0731  167.6868  158.6169  149.2270 
Results

The results for the uncontaminated case based on classic estimator are given in Table 5 as follows:
When the distribution is approximately symmetric (\(k_3=0.5\)), then the p of SC, WV and Shewhart method are comparable, while the SC method has a noticeable smaller p values. When the skewness increases, the ARL values decrease for all design schemes while the ARL values of the Shewhart chart decrase too much and are quite lower than others. The ARL values based on Shewhart and WV methods are lower than the SC method. So the SC method performs better than the others, especially for skewness. According to the p and ARL values, there is no difference between the Weibull, gamma and lognormal distributions. It is seen from the results, in the case of skewness, the Shewhart charts does not perform well any more. So we can recommend to use asymmetric control charts based on WV and SC methods (see more details in [13]).

The results for the contaminated case based on classic estimator are given in Table 6 as follows:
When we consider the contamination in the skewed distributed data, the WV and SC are effected so much from the outliers. So control charts based on WV and SC methods do not perform well any more. So we reccomend to use asymmetric control charts based on robust estimator.

The results for the uncontaminated case based on robust estimator are given in Table 7 as follows:
As the skewness increases, the MWV method gives better results than the MS, MSC gives better results than the MS and WV. The MSC method works very well for all skewed distributions for small and large sample sizes for all skewed distributions, except gamma distribution for \(n=10\).
When the skewed data are uncontaminated, the performance of the control charts based on WV and SC methods using classic estimators is comparable with the modified control charts based on MWV and MSC methods using robust estimators.

The results for the contaminated case based on robust estimator are given in Tables 8, 9 and 10 for Weibull, lognormal and gamma distributed data, respectively, as follows:
The p values for MSC method for gamma distribution are increasing when the number of the sample size is increasing. So the modified models perform well for large size, except MSC for gamma distribution. MSC method has the lowest p values and the highest ARL values for all skewed distributions in all designs. So this modified method has the best performance in the case of contamination in Phases I and II for the skewed data .
We investigate the effect of nonnormality on estimated limits under the contamination. The SC and MSC methods have the best performance for all design schemes, especially in the case of skewness.
Conclusion
Control charts are known to be effective tools for monitoring the quality of process and are applied in many industries. In this study, we consider the nonnormality and the contamination for the R control charts. We propose to use the interquartile range estimator of the standard deviation to modify the methods. We study the effect of the estimator on control chart performance under nonnormality for moderate sample size (30 subgroups of 5–10). To evaluate the control chart performance, we obtain p and ARL values of this control charts and the results used to compare the methods. We consider the design schemes where the Phase I and the Phase II data are noncontaminated and contaminated. The results are: The Shewhart chart has the worst performance for all design schemes, since the p values of the Shewhart chart are quite higher than others. As the skewness increases, the p values of the Shewhart chart increase too much and are effected by skewness. So the asymmetric control charts based on WV and SC methods can be used in the case of skewness. When the skewed data are uncontaminated, the performance of the control charts based on WV and SC methods using classic estimators is comparable with the modified control charts based on MWV and MSC methods using robust estimators. When there is no contamination, the SC and MSC methods work very well for all skewed distributions for small and large sample sizes for all skewed distributions. However, in the case of contamination, control charts based on WV and SC methods do not perform well any more. The MSC method has the lowest p values and the highest ARL values for all skewed distributions under contamination and so has the best performance. We reccomend to use asymmetric control charts based on MSC method for the skewed data in the case of contamination in Phases I and II.
As a future research, the proposed control chart can be extended using some other sampling schemes such as repetitive sampling, multiple dependent state sampling, ranked set sampling and neoteric rank set sampling.
As another future research, it is possible to consider other skewed distributions as heavytailed distributions.
Notes
Funding
This study was not funded.
Compliance with ethical standards
Conflict of interest
Derya Karagöz declares that she has no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
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