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Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process

  • Derya Karagöz
Open Access
Original Research
  • 79 Downloads

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 estimator 

Introduction

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. Abu-Shawiesh [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 non-normality 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 non-skew 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 non-skew 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. Wei-Heng 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 log-transformed 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 out-of-control 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 non-normality 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 non-normality 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 Weibull, gamma and lognormal distributions are chosen as skewed distributions since they can represent a wide variety of shapes from nearly symmetric to highly skewed.
  • The probability density function of the Weibull distribution is defined as
    $$\begin{aligned} f(x|\beta ,\lambda )=\beta \lambda ^\beta {x}^{\beta -1}\exp (-x\lambda )^\beta \end{aligned}$$
    for \(x>0\), where \(\beta\) is a shape parameter and \(\lambda\) is a scale parameter.
  • The probability density function of the gamma distribution is defined as
    $$\begin{aligned} f(x|\alpha ,\beta )=\frac{1}{\Gamma (\alpha )\beta ^\alpha }x^{\alpha -1}\exp \left( -\frac{x}{\beta }\right) \end{aligned}$$
    for \(x>0\), where \(\alpha\) is a shape parameter and \(\beta\) is a scale parameter.
  • The probability density function of the lognormal distribution is defined as
    $$\begin{aligned} f(x|\sigma ,\mu )=\frac{1}{x\sigma \sqrt{2\pi }}\exp \left( -\frac{({{\text{ln}}(x)-\mu })^2}{2\sigma ^2}\right) \end{aligned}$$
    for \(x>0\), where \(\sigma\) is a scale parameter and \(\mu\) is a location parameter.

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.

The first scale estimator is the mean of the sample range
$$\begin{aligned} \bar{R}= \frac{1}{k}\sum _{i=1}^{k}R_i \end{aligned}$$
(2.1)
where \(R_i\) is the range of the ith sample. An unbiased estimator of \(\sigma\) is \(\bar{R}/d_2(n)\). We also consider the mean of the sample interquartile ranges since the mean of the sample range is not robust against to outliers. The mean of the sample interquartile ranges (IQRs) is defined by
$$\begin{aligned} \bar{\text {IQR}}=\frac{1}{k}\sum _{i=1}^{k}{{\text {IQR}}_{i}} \end{aligned}$$
(2.2)
where IQR\(_{i}\) is the interquartile range of sample i
$$\begin{aligned} {\text {IQR}}_{i} = Q_{75,i}-Q_{25,i} \end{aligned}$$
(2.3)
where \(Q_{r,i}\) is the rth percentile of the values in sample i.

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 conventional control charts when the distribution is normal are the Shewhart control charts. We first consider the Shewhart method proposed by Montgomery [15]. The Shewhart R control chart limits are given as follows:
$$\begin{aligned} {\text {UCL}}_{\mathrm{{Shewhart}}}&= {} \left( 1+\frac{3d_3}{d_2}\right) \bar{R}, \end{aligned}$$
(2.4)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{Shewhart}}}&= {} \left( 1-\frac{3d_3}{d_2}\right) \bar{R} \end{aligned}$$
(2.5)
where \(d_2\) and \(d_3\) are constants that depend on the subgroup size n, and are calculated when the distribution is normal [15].
The MS R control chart limits are derived by replacing the range with the interquartile range as follows:
$$\begin{aligned} {\text {UCL}}_{\mathrm{{MS}}}&= {} \left( 1+\frac{3d_3^Q}{d_2^Q}\right) \bar{{\text {IQR}}}, \end{aligned}$$
(2.6)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{MS}}}&= {} \left( 1-\frac{3d_3^Q}{d_2^Q}\right) \bar{{\text {IQR}}} \end{aligned}$$
(2.7)
where \(d_2^Q\) and \(d_3^Q\) are constants that depend on the subgroup size n, and are calculated when the distribution is skewed.

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 \(\bar{R}\).

When the parameters of the process are unknown, the WV R control chart limits are defined by Bai and Choi [2] as follows:
$$\begin{aligned} {\text {UCL}}_{\mathrm{{WV}}}&= {} \bar{R}\left[ 1+3\frac{d_{3}^{*}}{d_{2}^{*}}\sqrt{2\hat{P}_{x}}\right] \end{aligned}$$
(2.8)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{WV}}}&= {} \bar{R}\left[ 1-3\frac{d_{3}^{*}}{d_{2}^{*}}\sqrt{2\left( 1-\hat{P}_{x}\right) }\right] \end{aligned}$$
(2.9)
where \(d_{2}^{*}\) and \(d_{3}^{*}\) are the control chart constants for R chart based on WV. These constants which are defined as the mean and standard deviation of relative range \(\left( \frac{R}{\sigma }\right)\) have been obtained under the non-normality assumption. These values can be computed via numerical integration once the distribution is specified. In Eq. (2.9\(P_{X}\) indicates the probability that can be estimated by using the number of observations less than or equal to
$$\begin{aligned} \bar{\bar{X}}: \hat{P}_{X}=\frac{\sum _{i=1}^k \sum _{j=1}^{n} \delta \left( \bar{\bar{X}}-X_{ij}\right) }{nk} \end{aligned}$$
(2.10)
where k and n are the number of samples and the number of observations in a subgroup, and \(\delta (X) =1\) for \(X\ge 0, 0\) otherwise. Usually, \(\mu _{x}\) is estimated by the grand mean of the subgroup means \(\bar{\bar{X}}\) and \(\mu _{R}\) is estimated by the mean of the subgroup ranges \(\bar{R}\) [2].
In this paper, we propose the MWV method in which the mean of the subgroup ranges is replaced by the mean of the subgroup interquartile ranges. If the parameters of the process are unknown, the MWV R control chart limits are given by
$$\begin{aligned} {\text {UCL}}_{\mathrm{{MWV}}}&= {} \bar{{\text {IQR}}}\left[ 1+3\frac{d_{3}^{Q}}{d_{2}^{Q }}\sqrt{2\hat{P}_{x}^Q}\right] \end{aligned}$$
(2.11)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{MWV}}}&= {} \bar{{\text {IQR}}}\left[ 1-3\frac{d_{3}^{Q}}{d_{2}^{Q }}\sqrt{2\left( 1-\hat{P}_{x}^Q\right) }\right] \end{aligned}$$
(2.12)
where \(d_{2}^{Q }\) and \(d_{3}^{Q }\) are the control chart constants of MWV R control charts. These constants which are defined as the mean and standard deviation of interquartile range \(\left( \frac{{\text {IQR}}}{\sigma }\right)\) have been obtained under the non-normality assumption, see in “Measuring estimator’s efficiency” and “Determination of control charts constants” sections. In this paper, this constant based on classic and robust estimators is obtained via simulation for each skewed distribution, because of the difficulty in numerical integration. Equation (2.12) allows the probability to be estimated from
$$\begin{aligned} \hat{P}_{X}^Q=\frac{\sum _{i=1}^k \sum _{j=1}^{n} \delta \left( \bar{{\text {TM}}_\alpha }-X_{ij}\right) }{nk} \end{aligned}$$
(2.13)
where k and n are the number of samples and the number of observations in a subgroup, respectively, and \(\delta (X) =1\) for \(X\ge 0, 0\) otherwise. In Eq. (2.13), \(\bar{{\text {TM}}_\alpha }\) is the mean of the sample trimmed means, defined by
$$\begin{aligned} \bar{{\text {TM}}_\alpha }=\frac{1}{k}\sum _{i=1}^k{\bar{{\text {TM}}_{(v)_i}}} \end{aligned}$$
(2.14)
where TM\(_{(v)_i}\) denotes the vth ordered value of the sample trimmed means defined by
$$\begin{aligned} \bar{{\text {TM}}_{(v)_i}}=\frac{1}{n-2 \lceil n \alpha \rceil } \left[ \sum _{j=\lceil n\alpha \rceil + 1}^{n-\lceil n\alpha \rceil }{X_{(ij)}}\right] \end{aligned}$$
(2.15)
where \(\alpha\) denotes the percentage of samples to be trimmed and \(\lceil n \alpha \rceil\) denotes the ceiling function, i.e., the smallest integer not less than \(n\alpha\).

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.

If the parameters of the process are unknown, the SC R control chart limits are defined by Chan and Cui [3] as follows:
$$\begin{aligned} {\text {UCL}}_{\mathrm{{SCR}}}&= {} \left[ 1+\left( 3+d_{4}^{*}\right) \frac{d_{3}^{*}}{d_{2}^{*}}\right] \bar{R} \end{aligned}$$
(2.16)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{SCR}}}&= {} \left[ 1+\left( -3+d_{4}^{*}\right) \frac{d_{3}^{*}}{d_{2}^{*}}\right] ^{+}\bar{R} \end{aligned}$$
(2.17)
where \(d_{4}^{*}\) is the control chart constant that is obtained as follows:
$$\begin{aligned} d_{4}^{*}=\frac{\frac{4}{3}k_{3}(R)}{1+0.2k_{3}^{2}(R)} \end{aligned}$$
(2.18)
where \(k_{3}(R)\) is the skewness of the subgroup range R [3].
In this paper, we propose MSC method in which the mean of the subgroup ranges is replaced by the mean of the subgroup interquartile ranges. If the parameters of the process are unknown, the MSC R control chart limits are defined as follows:
$$\begin{aligned} {\text {UCL}}_{\mathrm{{MSCR}}}&= {} \left[ 1+\left( 3+d_{4}^{Q }\right) \frac{d_{3}^{Q}}{d_{2}^{Q}}\right] \bar{{\text {IQR}}} \end{aligned}$$
(2.19)
$$\begin{aligned} {\text {LCL}}_{\mathrm{{MSCR}}}&= {} \left[ 1+\left( -3+d_{4}^{Q}\right) \frac{d_{3}^{Q }}{d_{2}^{Q }}\right] ^{+}\bar{{\text {IQR}}} \end{aligned}$$
(2.20)
where \(d_{4}^{Q }\) are the control chart constant which is obtained for the MSC method as follows:
$$\begin{aligned} d_{4}^{Q }=\frac{\frac{4}{3}k_{3}({\text {IQR}})}{1+0.2k_{3}^{2}({\text {IQR}})} \end{aligned}$$
(2.21)
where \(k_{3}\) (IQR) is the skewness of the subgroup interquartile ranges.

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

In this section, we evaluate the effect of outliers on the accuracy of the conventional and proposed robust estimators by means of a Monte Carlo simulation. \((M=50{,}000)\) simulation runs of 30 (\(k=30\)) subgroups each of size \(\hbox {n}=5,10\) are performed to generate data under the skewed distributions. The generated data are Weibull, lognormal and gamma distributions with different parameters as presented in Table 1. The process dispersion is estimated by both classic and robust methods. We consider four models in the case of no outliers and outliers like [9],
  • 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.

We thus allow that some observations come from a different skewed population, and in the last two models, we allow for the occurrence of gross errors.
Table 1

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

We run the simulation \(M=50{,}000\) times and generate \(k=30\) samples of size \(n=5,10\) according to different simulation schemes and compute the scale estimate \(\hat{\sigma }_j\) for each sample for \(j=1,\ldots ,M\). For each simulation setting and for estimators, we compute the RMSE of the scale estimator
$$\begin{aligned} {\text {RMSE}}_\sigma = \sqrt{\frac{1}{M}\sum _{j=1}^{M}(\hat{\sigma }_j-\sigma )^2}. \end{aligned}$$
where \(\hat{\sigma }_j\) is the robust estimation of the standart deviation \(\hat{\sigma }\).
Table 2

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

The results for Weibull, lognormal and gamma distributions are reported in Table 2. The conclusions from the study are as follows:
  1. (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.

     
  2. (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.

     
  3. (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.

     
  4. (iv)

    In the presence of outliers, the classic scale estimator has the highest RMSE of all skewed distributions.

     
  5. (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 non-normality 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).

In this paper, these constants based on the classic and robust estimators are obtain via simulation for each skewed distribution, because of the difficulty of numerical integration. These all constants are obtained for three skewed distributions via simulation. We obtain \(E(\bar{{\text {IQR}}})\) by simulation: we generate 100,000 times k samples of size n, compute IQR for each instance and take the average of the values. The results of the constants for the Shewhart, WV and SC methods are presented in Table 3 for \(k=30\) and \(n=5,10\). Moreover, the results of the constants for the MS, MWV and MSC methods are presented in Table 4 for \(k=30\) and \(n=5,10\).
Table 3

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

Table 4

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 two-phase procedure. In Phase I, control charts are used to define the in-control 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 in-control 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 the process control, the R, S and \(S^2\) control charts are widely used tools to monitor process variability. Let \(X_{ij}\) , \(i=1,2,3,\ldots\) and \(j=1,\ldots ,n\) denote independent random samples of size n taken in sequence on the process variable of interest; let \(\hat{\sigma }_i\) denote an estimate of the process standard deviation \(\sigma\) based on the ith sample. The control limits are
$$\begin{aligned} \hat{\text {UCL}}=U_n\hat{\sigma } \quad \quad \hat{\text {LCL}}=L_n\hat{\sigma } \end{aligned}$$
where \(U_n\) and \(L_n\) are chosen based on the skewness for this study so that the desired control chart limits are constructed when the process is in control. When the \(\hat{\sigma }_i\) falls with in the control limits, the process is called in control. Let \(E_i\) denote the event that the ith sample standard deviation is beyond the limits. Further, denote by \(P(Ei| \hat{\sigma })\) the conditional probability that is given for \(\hat{\sigma }\); the sample standard deviation \(\hat{\sigma _i}\) is beyond the control limits
$$\begin{aligned} P(Ei|\hat{\sigma })=P(\hat{\sigma _{i}} < {\text {LCL}} \quad or\quad \hat{\sigma _{i}} > {\text {UCL}}| \hat{\sigma } ) \end{aligned}$$
(3.1)
The RL as the run length is the number of subgroups until the first \(\hat{\sigma }_i\) falls beyond the limits. Given \(\hat{\sigma }\), when the \(E_s\) and \(E_t (s = t)\) are independent, and therefore, the distribution of the run length is geometric with parameter \(P(Ei|\hat{\sigma })\). The mean of the geometric distribution is given by 1 / p . Consequently, the conditional ARL is given by
$$\begin{aligned} E(RL| \hat{\sigma })= \frac{1}{P(Ei|\hat{\sigma })} \end{aligned}$$
(3.2)
When the standard deviation is estimated, the conditional runlength—the run length given an estiamte of \(\sigma\)—has a geometric distribution. However, the unconditional run length distribution the run length distribution averaged overall possible values of the estimated \(\sigma\)—is not geometric [20].
In contrast with the conditional RL distirbution, the marginal RL distribution takes into account the random variability introduced into the charting procedure through parameter estimation. It can be obtained by averaging the conditional RL distribution over all possible values of the parameter estimates. The unconditional p and unconditional average run length are given in [19] as, respectively
$$\begin{aligned} p&= {} E(P(Ei|\hat{\sigma })) \end{aligned}$$
(3.3)
$$\begin{aligned} {\text {ARL}}&= {} E(E({\text {RL}}| \hat{\sigma }))=E\left( \frac{1}{P(Ei|\hat{\sigma })}\right) . \end{aligned}$$
(3.4)
These expectations are simulated by generating 10,000 times k data samples of size n: numerous datasets are generated from the contaminated skewed distributions and computing for each data set the conditional value \((Ei|\hat{\sigma })\). By averaging these values, we obtain the unconditional values over the data sets. Note that for the calculation of the control limits in Phase I the process is considered to be in control [18].

In this section, we consider design schemes for the R control chart for non-contaminated 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 two-stage 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 real-time process monitoring [21].

The simulation consists of two phases is run by using MATLAB R2013. The steps of each phase are described as follows.

Phase I:
  1. 1.a.

    Generate n i.i.d. Weibull \((\beta ,1)\), gamma \((\alpha ,1)\) and lognormal \((0,\sigma )\) varieties for \(n=3,5,7,10\).

     
  2. 1.b.

    Repeat step 1.a 30 times \(\left( k=30\right) .\)

     
  3. 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.

     
Phase II:
  1. 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. 2.b.

    Repeat step 2.a 100 times (\(k=100\)).

     
  3. 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.

     
  4. 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.

     
  5. 2.e.

    Repeat steps 1.a through 2.d, 100.000 times and obtain p and ARL values for each method.

     
In the simulation study, we consider non-contaminated and contaminated data set in Phases I and II. We consider the \(20\%\) trimmed mean, which trims the six smallest and the six largest sample trimmed means when \(k=30\).
  • Non-contaminated 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.

The simulation results of p and ARL for the R control chart for non-contaminated data under skewed distributions are given in Tables 5 and 7. The results of p and ARL for the R control chart for contaminated Weibull, lognormal and gamma distrubuted data are given in Tables 8, 9 and 10, respectively.
Table 5

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

Table 6

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

*ARL values of Weibull multiplies with \(1.0e+04\) for \(n=5, 1.0e+05\) for \(n=10\)

*ARL values of lognormal multiplies with \(1.0e+6\) for \(n=5\) and \(1.0e+05\) for \(n=10\)

*ARL values of gamma multiplies with \(1.0e+05\) for \(n=5\) and \(1.0e+04\) for \(n=10\)

Table 7

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

Table 8

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

*ARL values multiplies with \(1.0e+03\)

Table 9

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

Table 10

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

*ARL values multiplies with \(1.0e+04\)

Results

In this section, the performance of design schemes is evaluated. When the process in control, it is expected that p is to be as low as possible and ARL is to be as high as possible. First we consider the design scheme where the process follows skewed distribution and the Phase I data are non-contaminated. Tables 5 and 7 present the p and the ARL values for the R control chart based on classic and robust estimators under the skewed distributions. The tables shows that :
  • 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 .

When the simulation program is run for \(n=25\), the results are the same results as \(n=10\). So we can say that the results are same for large sample size.

We investigate the effect of non-normality 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 non-normality 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 non-normality 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 non-contaminated 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 heavy-tailed 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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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of StatisticsHacettepe UniversityBeytepe, AnkaraTurkey

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