# Monitoring Coefficient of Variation using One-Sided Run Rules control charts
in the presence of Measurement Errors

Phuong Hanh Trana,b and Cédric Heuchenneb and Huu Du Nguyena Corresponding
author. Phuong Hanh Tran. Email<EMAIL_ADDRESS>aDivision of Artificial
Intelligence, Dong A University, Danang, Vietnam; bHEC Liège - Management
School of the University of Liège, Liège, Belgium

###### Abstract

We investigate in this paper the effect of the measurement error on the
performance of Run Rules control charts monitoring the coefficient of
variation (CV) squared. The previous Run Rules CV chart in the literature is
improved slightly by monitoring the CV squared using two one-sided Run Rules
charts instead of monitoring the CV itself using a two-sided chart. The
numerical results show that this improvement gives better performance in
detecting process shifts. Moreover, we will show through simulation that the
precision and accuracy errors do have negative effect on the performance of
the proposed Run Rules charts. We also find out that taking multiple
measurements per item is not an effective way to reduce these negative
effects.

###### keywords:

Run Rules chart, Markov chain, Coefficient of Variation, Measurement Errors.

††articletype: ARTICLE

##  1 Introduction

Among important statistical characteristics of a variable, the coefficient of
variation (CV) is widely used to evaluate the stability or concentration of
the random variable around the mean. It is defined as the ratio between the
standard deviation to the mean, $\gamma=\sigma/\mu$. In many industrial
processes, keeping the value of this coefficient of a characteristic of
interest within the permissible range means assuring the quality of products.
A number of examples have been illustrated in the literature for the
applications of the CV in industry. [5] presented an example where the quality
of interest is the pressure test drop time from a sintering process
manufacturing mechanical parts. In this example, the presence of a constant
proportionality between the standard deviation of the pressure drop time and
its mean was confirmed. The CV was then monitored to detect changes in the
process variability. [16] showed that it is useful to monitor the CV in
detecting the presence of chatter, a severe form of self-excited vibration in
the machining process which leads to many machining problems. More examples
about the need of using the CV as a measure of interest has been discussed in
[7]. Because of its wide range of applications, monitoring the CV is a major
objective in many studies in statistical process control, see, for example,
[3],[4], [17], and [6].

Along with the development of the advanced control charts monitoring the CV
with improved performance, recent researches are paying attention to the
effect of the measurement error on the CV control chart. This makes these
researches become more in touch reality since the measurement error is often
present in practice. A Shewhart control chart monitoring the CV under the
presence of measurement error (ME) was suggested by [18]. [12] improved the
linear covariate error model for the CV in [18] and then proposed the EWMA CV
control chart with ME. Very recently, [8] and [14] studied the effect of ME on
the variable sampling interval control chart and the cumulative sum control
chart monitoring the CV, respectively.

One of the reasons that leads to the introduction of many advanced control
charts monitoring the CV is to overcome a drawback of the Shewhart CV chart
that it is not sensitive to the large shifts. However, the Shewhart chart is
still popularly used thanks to its simplicity in implementation. From this
point of view, the Run Rules charts are advantageous: they are easy to
implement (compared to, for example, the EWMA control chart or the CUSUM
control chart, even these charts may bring better performance) and they can
improve remarkably the performance of the Shewhart chart in detecting small or
moderate process shifts. The aim of this paper is to investigate the
performance of Run Rules CV control chart under the presence of ME. In fact,
the Run Rules chart monitoring the CV has been studied in [2]. However, the ME
has not been considered. Moreover, in this study the authors only focused on
the two-sided charts (the one-sided chart has been mentioned, but quite
sketchily without explanation for the design) with the CV monitored directly.
We improve this design by monitoring the CV squared and presenting the design
of the two one-sided Run Rules charts in detail.

The paper consists of eight sections and is organized as follows. Followed by
the introduction in Section 1, Section 2 presents a brief review of the
distribution of the sample coefficient of variation. The design and the
implementation of two one-sided Run Rules control charts monitoring the CV
squared (denoted as RR${}_{r,s}-{\gamma}^{2}$ charts) are presented in section
3. Section 4 is for the performance of these charts. A linear covariate error
model for the CV is reintroduced in section 5. The design of control charts in
the presence of measurement errors and the effect of the measurement error on
the RR${}_{r,s}-{\gamma}^{2}$ charts are displayed in section 6. Section 7 is
devoted to an illustrative example. Some concluding remarks are given in
section 8 to conclude.

##  2 A brief review of distribution of the sample coefficient of variation

In this section, the distribution of the CV is briefly presented. The CV of a
random variable $X$, say $\gamma$, is defined as the ratio of the standard
deviation $\sigma=\sigma(X)$ to the mean $\mu=E(X)$; i.e.,

$\gamma=\frac{\sigma}{\mu}.$

Suppose that a sample of size $n$ of normal i.i.d. random variables
$\\{X_{1},\ldots,X_{n}\\}$ is collected. Let $\bar{X}$ and $S$ be the sample
mean and the sample standard deviation of these variables, i.e.,

$\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}$

and

$S=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}}.$

Then the sample coefficient of variation $\hat{\gamma}$ of these variables is
defined as

$\hat{\gamma}=\frac{S}{\bar{X}}.$

The probability distribution of the sample CV $\hat{\gamma}$ has been studied
in the literature by many authors. However, the exact distribution of
$\hat{\gamma}$ has a complicated form. The approximate distribution is then
widely used as an alternative. In this study, we apply the following
approximation of $F_{\hat{\gamma}}(x|n,\gamma)$, the $c.d.f$ (cumulative
distribution function) of $\hat{\gamma}$, which is suggested by [5]:

$F_{\hat{\gamma}}(x|n,\gamma)=1-F_{t}\left(\left.\frac{\sqrt{n}}{x}\right|n-1,\frac{\sqrt{n}}{\gamma}\right),$
(1)

where $F_{t}\left(.\left|n-1,\frac{\sqrt{n}}{\gamma}\right.\right)$ is the
c.d.f. of the noncentral $t$ distribution with $n-1$ degrees of freedom and
noncentrality parameter. This approximation is only sufficiently precise when
$\gamma<0.5$. This condition is in general satisfied in our case as it is very
frequent that the CV takes small values to ensure the stability of a process.
More details on this problem have been discussed in [12].

For the case of the sample CV squared ($\hat{\gamma}^{2}$), Castagliola et al.
[5] showed that $\frac{n}{\hat{\gamma}^{2}}$ follows a noncentral $F$
distribution with $(1,n-1)$ degrees of freedom and noncentrality parameter
$\frac{n}{\gamma^{2}}$. Then, they deduced the $c.d.f$
$F_{\hat{\gamma}^{2}}(x|n,\gamma)$ of $\hat{\gamma}^{2}$ as

$F_{\hat{\gamma}^{2}}(x|n,\gamma)=1-F_{F}\left(\left.\frac{n}{x}\right|1,n-1,\frac{n}{\gamma^{2}}\right),$
(2)

where $F_{F}\left(.\left|1,n-1,\frac{n}{\gamma^{2}}\right.\right)$ is the
$c.d.f$ of the noncentral $F$ distribution. The corresponding density function
of $\hat{\gamma}^{2}$ is then

$f_{\hat{\gamma}^{2}}(x|n,\gamma)=\frac{n}{x^{2}}f_{F}\left(\left.\frac{n}{x}\right|1,n-1,\frac{n}{\gamma^{2}}\right),$
(3)

where $f_{F}\left(.\left|1,n-1,\frac{n}{\gamma^{2}}\right.\right)$ is the
density function of the noncentral $F$ distribution

Figure 1 presents the density distribution of $\gamma^{2}$ for $n=5$ and some
different values of $\gamma_{0}$.

PLEASE INSERT FIGURE 1 HERE

##  3 Design and implementation of the RR${}_{r,s}-{\gamma}^{2}$ control chart

In the literature, the Run Rules control charts monitoring the CV has been
investigated in [2] with two-sided charts. However, since the distribution of
$\gamma^{2}$ is asymmetric (as can be seen from the equation (2) and also from
Figure 1), these two-sided charts lead to the problem of $ARL$-biased (Average
Run Length) performance, i.e. the out-of-control $ARL_{1}$ values are larger
than the in-control values $ARL_{0}$. This problem was also pointed out in
[2]. Therefore, we overcome this $ARL$-biased property by designing
simultaneously two one-sided charts to detect both the increase and the
decrease at the CV squared. In particular, we suggest defining two one-sided
Run Rules control charts monitoring the CV squared, involving:

  * •

a lower-sided $r$-out-of-$s$ Run Rules control chart (denoted as
RR${}_{r,s}^{-}-{\gamma}^{2}$) to detect a decrease in $\hat{\gamma}_{i}$ with
a lower control limit
$LCL^{-}=\mu_{0}(\hat{\gamma}^{2})-k_{d}.\sigma_{0}(\hat{\gamma}^{2})$ and a
corresponding upper control limit $UCL^{-}=+\infty$,

  * •

an upper-sided $r$-out-of-$s$ Run Rules control chart (denoted as
RR${}_{r,s}^{+}-{\gamma}^{2}$) to detect a decrease in $\hat{\gamma}_{i}$ with
a lower control limit
$UCL^{+}=\mu_{0}(\hat{\gamma}^{2})+k_{u}.\sigma_{0}(\hat{\gamma}^{2})$ and a
corresponding lower control limit $LCL^{+}=-\infty$,

where $k_{d}>0$ and $k_{u}>0$ are the chart parameters of the
RR${}_{r,s}^{-}-{\gamma}^{2}$ and RR${}_{r,s}^{+}-{\gamma}^{2}$ charts,
respectively.

The closed forms of $\mu_{0}(\hat{\gamma}^{2})$ and
$\sigma_{0}(\hat{\gamma}^{2})$ have not been presented in the literature. We
apply in this study the accurate approximations provided by Breunig [1] for
both $\mu_{0}(\hat{\gamma}^{2})$ and $\sigma_{0}(\hat{\gamma}^{2})$ as
follows:

$\displaystyle\mu_{0}(\hat{\gamma}^{2})$ $\displaystyle=$
$\displaystyle\gamma_{0}^{2}\bigg{(}1-\frac{3\gamma_{0}^{2}}{n}\bigg{)},$ (4)
$\displaystyle\sigma_{0}(\hat{\gamma}^{2})$ $\displaystyle=$
$\displaystyle\sqrt{\gamma_{0}^{4}\left(\frac{2}{n-1}+\gamma_{0}^{2}\left(\frac{4}{n}+\frac{20}{n(n-1)}+\frac{75\gamma_{0}^{2}}{n^{2}}\right)\right)-(\mu_{0}(\hat{\gamma}^{2})-\gamma_{0}^{2})^{2}}.$
(5)

Given the value of the control limit for each chart, an out-of-control signal
is given at time $i$ if $r$-out-of-$s$ consecutive $\hat{\gamma}_{i}$ values
are plotted outside the control interval, i.e. $\hat{\gamma}_{i}^{2}<LCL^{-}$
in the lower-sided chart and $\hat{\gamma}_{i}^{2}>UCL^{+}$ in the upper-sided
chart. The control charts designed above is called $pure$ Run Rules type
chart. In this study, we only consider the 2-out-of-3, 3-out-of-4 and 4-out-
of-5 Run Rules charts. The performance of the proposed charts is measured by
the $ARL$ which is calculated by using Markov chain as follows.

Firstly, we define the matrix $\mathbf{P}$ of the embedded Markov chain. For
the two one-sided RR${}_{2,3}-{\gamma}^{2}$ control charts, $\mathbf{P}$ is
defined by

$\mathbf{P}=\left(\begin{array}[]{cc}\mathbf{Q}&\mathbf{r}\\\ &\\\
\mathbf{0}^{T}&1\end{array}\right)=\left(\begin{array}[]{ccc|c}0&0&p&1-p\\\
p&0&0&1-p\\\ 0&1-p&p&0\\\ \hline\cr 0&0&0&1\end{array}\right),$ (6)

where $\mathbf{Q}$ is a $(3,3)$ matrix of transient probabilities,
$\mathbf{r}$ is a $(3,1)$ vector satisfied
$\mathbf{r}=\mathbf{1}-\mathbf{Q}\mathbf{1}$ with $\mathbf{1}=(1,1,1)^{T}$ and
$\mathbf{0}=(0,0,0)^{T}$, $p$ is the probability that a sample drops into the
control interval. The corresponding $(3,1)$ vector $\mathbf{q}$ of initial
probabilities associated with the transient states is
$\mathbf{q}=(0,0,1)^{T}$, i.e. the third state is the initial state.

For the case of RR${}_{3,4}-{\gamma}^{2}$ control charts, the transient
probability matrix $\mathbf{Q}_{(7\times 7)}$ is given by

$\mathbf{Q}=\left(\begin{array}[]{ccccccc}0&0&p&0&0&0&0\\\ 0&0&0&0&p&0&0\\\
0&0&0&0&0&1-p&p\\\ p&0&0&0&0&0&0\\\ 0&1-p&p&0&0&0&0\\\ 0&0&0&1-p&p&0&0\\\
0&0&0&0&0&1-p&p\\\ \end{array}\right).$ (7)

In this case, the seventh state in the vector $\mathbf{q}=(0,0,0,0,0,0,1)^{T}$
is the initial state.

Extended to “longer” $(4,5)$ Run Rules, the $(15,15)$ matrix $\mathbf{Q}$ of
transient probabilities for the two one-sided RR${}_{4,5}-\gamma^{2}$ control
charts is

$\small\mathbf{Q}=\left(\begin{array}[]{ccccccccccccccc}1-p&p&0&0&0&0&0&p&0&0&0&0&0&0&0\\\
0&0&p&1-p&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&p&1-p&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1-p&p&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&1-p\\\
0&0&0&0&0&0&0&0&0&0&0&0&p&1-p&0\\\ 0&0&0&0&0&0&0&0&0&0&0&p&1-p&0&0\\\
0&0&$0$&0&0&0&0&1-p&p&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1-p&p&0&0&0&0&0&0&0\\\
0&0&0&0&p&1-p&0&0&0&0&0&0&0&0&0\\\ 0&0&p&1-p&0&0&0&0&0&0&0&0&0&0&0\\\
1-p&p&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1-p&p&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&p&1-p&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&p&1-p\\\
\end{array}\right),$ (8)

that corresponds to the $(15,1)$ initial probabilites vector
$\mathbf{q}=(0,\ldots,0,1)^{T}$ (i.e. the initial state is the 15th one).
These transient probability matrices has been presented in, for example, [11],
[10], [13].

Let us now suppose that the occurrence of an unexpected condition shifts the
in-control CV value $\gamma_{0}$ to the out-of-control value
$\gamma_{1}=\tau\times\gamma_{0}$, where $\tau>0$ is the shift size. Values of
$\tau\in(0,1)$ correspond to a decrease of the $\gamma_{0}$, while values of
$\tau>1$ correspond to an increase of $\gamma_{0}$. Then, the probability $p$
is defined by

  * •

for the RR${}_{r,s}^{-}-{\gamma}^{2}$ chart:

$p=P(\hat{\gamma}_{i}^{2}\geq
LCL^{-})=1-F_{\hat{\gamma}^{2}}(LCL^{-}|n,\gamma_{1}),$ (9)

  * •

for the RR${}_{r,s}^{+}-{\gamma}^{2}$ chart:

$p=P(\hat{\gamma}_{i}^{2}\leq
UCL^{+})=F_{\hat{\gamma}^{2}}(UCL^{+}|n,\gamma_{1}),$ (10)

where $F_{\hat{\gamma}^{2}}$ is defined in (2).

Once the matrix $\mathbf{Q}$ and the vector $\mathbf{q}$ have been determined,
the $ARL$ and the $SDRL$ (standard deviation of run length) are calculated by

$\displaystyle ARL$ $\displaystyle=$
$\displaystyle\mathbf{q}^{T}(\mathbf{I}-\mathbf{Q})^{-1}\mathbf{1},$ (11)
$\displaystyle SDRL$ $\displaystyle=$
$\displaystyle\sqrt{2\mathbf{q}^{T}(\mathbf{I}-\mathbf{Q})^{-2}\mathbf{Q}\mathbf{1}-ARL^{2}+ARL}.$
(12)

It is customary that a control chart is considered to be better than its
competitors if it gives a smaller value of the $ARL_{1}$ while the $ARL_{0}$
is the same. Therefore, the parameters of the RR${}_{r,s}-{\gamma}^{2}$
control charts should be the solution of the following equations:

  * •

for the RR${}_{r,s}^{-}-{\gamma}^{2}$ chart:

$ARL(k_{d},n,p,\gamma_{0},\tau=1)=ARL_{0},$ (13)

  * •

for the RR${}_{r,s}^{+}-{\gamma}^{2}$ chart:

$ARL(k_{u},n,p,\gamma_{0},\tau=1)=ARL_{0},$ (14)

where $ARL_{0}$ is predefined.

##  4 Performance of RR${}_{r,s}-{\gamma}^{2}$ control charts

Assigning the in-control value $ARL_{0}$ at $ARL_{0}=370.4$, the parameters
$k_{d}$ and $k_{u}$ of the lower-sided and upper-sided
RR${}_{r,s}-{\gamma}^{2}$ charts for some combinations of
$n\in\\{5,15\\},\gamma_{0}\in\\{0.05,0.1,0.2\\}$ are presented in Table 2.
Table 3 shows the corresponding $ARL_{1}$ values of the proposed charts for
various situations of the shift size $\tau$. The obtained results show that
the two one-sided RR${}_{r,s}-{\gamma}^{2}$ charts not only overcome the
$ARL$-biased problem (as the $ARL_{1}$ values are always smaller than the
$ARL_{0}$) but also outperform the two-sided RR-$\gamma$ charts investigated
in [2]. For example, with $\gamma_{0}=0.05,\tau=1.10$ and $n=5$ in the
RR${}_{2,3}-{\gamma}^{2}$ chart, we have $ARL_{1}=95.9$ ( Table 3 in this
study), which is smaller than $ARL_{1}=101.6$ (Table 2 in [2]).

INSERT TABLE 1 ABOUT HERE

INSERT TABLE 2 ABOUT HERE

INSERT TABLE 3 ABOUT HERE

##  5 Linear covariate error model for the coefficient of variation

The previous design for the RR${}_{r,s}-{\gamma}^{2}$ control charts is based
on a latent assumption that the values in the collected sample are measured
exactly without the measurement error. This assumption, however, is usually
not reached in practice and it is difficult to avoid the measurement error.
That leads to many authors have conducted their studies based on the
measurement error presence, see for instance [8, 15]

In this section, we suppose a linear covariate error model to the measurement
error, which is suggested by [9].

Suppose that the quality characteristic $X$ of $n$ consecutive items at step
ith is $(X_{i,1},X_{i,2},..,X_{i,n})$, where $X_{i,j}\sim
N(\mu_{0}+a\sigma_{0},b^{2}\sigma_{0}^{2})$ where $\mu_{0}$ and $\sigma_{0}$
are the in-control mean and standard deviation of $X$ and they are
independent. The parameters $a$ and $b$ represent the standardized mean and
standard deviation shift. The process has shifted if the process mean
$\mu_{0}$ and/or the process standard deviation $\sigma_{0}$ have changed
($a\neq 0$ and/or $b\neq 1$). Due to the measurement error, we only observe
the values $(X_{i,j,1}^{*},...,X_{i,j,m}^{*})$ of a set of $m$ measurement
operations instead the true values $X_{i,j}$. According to the linear
covariate error model, we assume
$X_{i,j,k}^{*}=A+BX_{i,j}+\varepsilon_{i,j,k}$, where $A$ and $B$ are two
known constants and $\varepsilon_{i,j,k}$ is a normal random error term with
parameters $(0,\Sigma_{M})$ and independent of $X_{i,j}$.

Let $\bar{X}_{i,j}^{*}$ denote the mean of $m$ observed quantities of the same
item $j$ at the ith sampling. It is straightforward to show that

$\bar{X}_{i,j}^{*}~{}\sim~{}N(\mu^{*},\sigma^{*2})=N\left(A+B(\mu_{0}+a\sigma_{0}),B^{2}b^{2}\sigma^{2}_{0}+\frac{\sigma^{2}_{M}}{m}\right).$

[12] showed that the CV of the quantity $\bar{X}_{i,j}^{*}$ is

$\gamma^{*}=\frac{\sigma^{*}}{\mu^{*}}=\frac{\sqrt{B^{2}b^{2}+\frac{\eta^{2}}{m}}}{\theta+B(1+a\gamma_{0})}\times\gamma_{0}.$
(15)

where
$\gamma_{0}=\frac{\sigma_{0}}{\mu_{0}},\eta=\frac{\sigma_{M}}{\sigma_{0}}$ and
$\theta=\frac{A}{\mu_{0}}$ are the in-control value of CV, the precision error
ratio and the accuracy error, respectively. The sample coefficient of
variation $\hat{\gamma}^{*}_{i}$ is defined by
$\hat{\gamma}^{*}_{i}=\frac{S_{i}^{*}}{\bar{\bar{X}}_{i}^{*}}$ where
$\bar{\bar{X}}_{i}^{*}$ and $S_{i}^{*}$ are the sample mean and the sample
standard deviation of $\bar{X}_{1,j}^{*},\ldots,\bar{X}_{n,j}^{*}$. The
$c.d.f$ of $\hat{\gamma}^{*2}$ can be obtained from (2) by simply replacing
$\gamma$ by $\gamma^{*}$, i.e., the $c.d.f$
$F_{\hat{\gamma}^{*2}}(x|n,\gamma^{*})$ of $\hat{\gamma}^{*2}$ is given by

$F_{\hat{\gamma}^{*2}}(x|n,\gamma^{*})=1-F_{F}\left(\frac{n}{x}\left|1,n-1,\frac{n}{\gamma^{*2}}\right.\right)$
(16)

##  6 Implementation and the performance of the RR${}_{r,s}-{\gamma}^{2}$
charts with measurement errors

Under the presence of measurement errors, the values
$\mu_{0}(\hat{\gamma}^{*2})$ and $\sigma_{0}(\hat{\gamma}^{*2})$ are
calculated as in (4) and (5), where $\gamma_{0}$ is replaced by
$\gamma_{0}^{*}$, which is defined from (15) with $a=0$ and $b=1$:

$\gamma^{*}_{0}=\frac{\sqrt{B^{2}+\frac{\eta^{2}}{m}}}{\theta+B)}\times\gamma_{0}.$
(17)

Suppose that the in-control value $\gamma_{0}$ is shifted to the out-of-
control value $\gamma_{1}$ with the size $\tau$, we can represent $\tau$
according to $a$ and $b$ as $\tau=b/(1+a\gamma_{0})$. Therefore, the out-of-
control CV of the observed quantity $\bar{X}^{*}_{i,j}$ can be expressed by

$\gamma^{*}_{1}=\frac{\sqrt{B^{2}b^{2}+\frac{\eta^{2}}{m}}}{\theta+\frac{Bb}{\tau}}\times\gamma_{0}.$
(18)

In the implementation of RR${}_{r,s}-{\gamma}^{2}$ control charts, the control
limits,
$UCL^{*+}=\mu_{0}(\hat{\gamma}^{*2})+k_{u}^{*}.\sigma_{0}(\hat{\gamma}^{*2})$
and
$LCL^{*-}=\mu_{0}(\hat{\gamma}^{*2})-k_{d}^{*}.\sigma_{0}(\hat{\gamma}^{*2})$,
are also found by solving the chart parameters $k_{d}$ and $k_{u}$ as the
solution of the following equations

  * •

for the RR${}_{r,s}^{-}-{\gamma}^{2}$ chart:

$ARL(k_{d},n,p,\gamma_{0},\theta,\eta,m,B,b)=ARL_{0},$ (19)

  * •

for the RR${}_{r,s}^{+}-{\gamma}^{2}$ chart:

$ARL(k_{u},n,p,\gamma_{0},\theta,\eta,m,B,b)=ARL_{0}.$ (20)

The $ARL$ in (19) and (20) should be calculated with the transition
probability matrix $\mathbf{Q}$ where the transition probability $p$ is
defined from (9) and (10) but with the $c.d.f$
$F_{\hat{\gamma}^{*2}}(x|n,\gamma^{*})$ of $\hat{\gamma}^{*2}$ in (16) instead
of $c.d.f$ $F_{\hat{\gamma}^{2}}$ in (2).

To investigate the performance of the RR${}_{r,s}-{\gamma}^{2}$ charts under
the appearance of the measurement error, we consider several possible values
of the parameters: $n\in\\{5,15\\}$, $\gamma_{0}\in\\{0.05,0.1,0.2\\}$,
$\eta\in\\{0,0.1,0.2,0.3,0.5,1\\}$,
$\theta\in\\{0,0.01,0.02,0.03,0.04,0.05\\}$, $m\in\\{1,3,5,7,10\\}$ and
$B\in\\{0.8,0.9,1,1.1,1.2\\}$. Without loss of generality, we assume in the
remaining that $b=1$. The in-control value CV is also set at $ARL_{0}=370.4$.

The control limits of the proposed charts for some specific values of these
parameters have been presented in Table 1. The other values of the control
limits for other situations of these parameters are not presented here but are
available upon request from authors.

Tables 4-7 show the corresponding values of the $ARL_{1}$ under different
effects of the parameters $\eta,\theta,m$ and $B$ of the linear covariate
model. Some simple conclusions can be drawn from these tables as follows.

  * •

The increase of the precision error ratio $\eta$ leads to an increase of the
$ARL_{1}$. However, this increase in the $ARL_{1}$ following the change of
$\eta$ is not significant, especially when $\eta\leqslant 0.3$. For example,
for the RR${}_{2,3}-\gamma^{2}$ chart with
$n=5,\gamma_{0}=0.05,B=1,m=1,\theta=0.05$ and $\tau=0.8$, we have
$ARL_{1}=93.12$ when $\eta=0.0$ and $ARL_{1}=93.20$ when $\eta=0.3$ (Table 4).
That means the precision error ratio does not affect much the performance of
the proposed charts.

  * •

The accuracy error $\theta$ has a negative impact on the
RR${}_{r,s}-\gamma^{2}$ charts’ performance: the larger the accuracy error
$\theta$ is, the larger the value $ARL_{1}$ is, i.e. the lower of the control
chart is in detecting the out-of-control condition. For example, in the
RR${}_{3,4}-\gamma^{2}$ chart with $n=5,\gamma_{0}=0.1,B=1,m=1,\eta=0.28$ and
$\tau=1.3$, we have $ARL_{1}=26.56$ when $\theta=0.0$ and $ARL_{1}=29.19$ when
$\theta=0.5$ (Table 5)

  * •

Given the value of other parameters, the variation of $B$ significantly
affects the performance of the RR${}_{r,s}-\gamma^{2}$ charts. For instance,
in Table 6 with the RR${}^{-}_{4,5}-\gamma^{2}$ control chart and
$n=5,m=1,\gamma_{0}=0.2,\eta=0.28,\theta=0.05,\tau=0.7$ we have
$ARL_{1}=14.43$ when $B=0.8$ and $ARL_{1}=13.93$ when $B=1.2$.

  * •

In many situations, taking multiple measurements per item in each sample is an
alternative to compensate for the effect of the measurement error. However,
the obtained results in this study show that this is not effective way. When
$m$ increases from $m=1$ to $m=10$, the $ARL_{1}$ decreases trivially or is
almost unchanged. For example, with
$n=5,B=1,\gamma_{0}=0.05,\eta=0.28,\theta=0.05,\tau=0.8$ in the
RR${}^{+}_{2,3}-\gamma^{2}$ we have $ARL_{1}=9.07$ for both $m=1$ and $m=10$
(Table 7).

  * •

In general, the RR${}_{r,s}-\gamma^{2}$ control charts give better performance
in detecting the small process shifts compared to the VSI-$\gamma^{2}$ control
chart investigated in [8], under the same condition of measurement errors. For
example, with the same values of
$n=5,\gamma_{0}=0.05,\eta=0.28,\theta=0.05,\tau=0.8$, we have $ARL_{1}=46.80$
for the RR${}_{4,5}-\gamma^{2}$ (Table 5 in this study), which is smaller than
$ARL_{1}=61.99$ for the VSI $\gamma^{2}$ control chart with
$(h_{S},h_{L})=0.1,4.0$ (Table 10 in [8]).

INSERT TABLE 4 ABOUT HERE

INSERT TABLE 5 ABOUT HERE

INSERT TABLE 6 ABOUT HERE

INSERT TABLE 7 ABOUT HERE

In practice, quality practitioners often prefer detecting a range of shifts
$\Omega=[a;b]$ since it is difficult to guess an exact value for the process
shift. In such situations, the statistical performance of the control chart
can be evaluated through the $EARL$ (expected average run length) defined as

$EARL=\int_{\Omega}ARL\times f_{\tau}(\tau)d\tau,$ (21)

where $f_{\tau}(\tau)$ is the distribution of process shift $\tau$ and $ARL$
is defined in (11). Without any information about $\tau$, one can choose the
uniform distribution in $\Omega$, i.e, $f_{\tau}(\tau)=\frac{1}{b-a}$.

The chart parameters are now defined as

  * •

for the RR${}_{r,s}^{-}-{\gamma}^{2}$ chart:

$EARL(LCL^{*-},n,p,\gamma_{0},\theta,\eta,m,B)=ARL_{0},$ (22)

  * •

for the RR${}_{r,s}^{+}-{\gamma}^{2}$ chart:

$EARL(UCL^{*+},n,p,\gamma_{0},\theta,\eta,m,B)=ARL_{0}.$ (23)

In the following simulation, we consider a specific range of decreasing shifts
$\Omega_{D}=[0.5,1)$ and increasing shifts $\Omega_{I}=(1,2]$. Figure 2-3 show
the change of $EARL$ of the RR-$\gamma^{2}$ control charts when $\eta$ varies
in $[0,1]$ and $\theta$ varies in $[0,0.05]$ for $\gamma_{0}=0.05$ and
$\gamma_{0}=0.2$, respectively. The slope of the plane which represents the
$EARL$ values from right to left and from outside to inside shows that the
larger the values of $\eta$ and $\theta$, the larger the value of $EARL$. That
is to say, these errors have negative effects on the performance of the
RR-$\gamma^{2}$ charts. For example, in Figure 2 when $n=5,B=m=1$, and
$\gamma=0.05$, we have $EARL=82.27$ for $\theta=\eta=0$ (corresponding to no
measurement errors), but $EARL=82.81$ for $\eta=0,\theta=0.05$ (corresponding
to the negative effect of accuracy error), $EARL=83.42$ for
$\theta=0,\eta=0.3$ (corresponding to the negative effect of precision error),
and $EARL=84.49$ for $\theta=0.05,\eta=0.5$ (corresponding to the negative
effect of both precision and accuracy error). The effect of $B$ and $m$ on the
$EARL$ is displayed in Figures 4-7 for both $\gamma_{0}=0.05$ and
$\gamma_{0}=0.2$. We obtain a similar trend as the case of the specific shift
size: When $B$ increases, the $EARL$ decreases and the $EARL$ does not change
significantly when $m$ increases. The almost constant $EARL$ line shows that
the effect of $m$ on these chart performance is insignificant. That is to say,
increasing the value of $m$ does not reduce the negative effect of measurement
errors on the charts. In contrast, the plot of the $EARL$ corresponding to
$n=15$ is always below the plot of the $EARL$ corresponding to $n=5$. That
means, the sample size has a great impact on the RR${}_{r,s}-\gamma^{2}$
charts’ performance regardless of the measurement error.

PLEASE INSERT FIGURE 2 HERE

PLEASE INSERT FIGURE 3 HERE

PLEASE INSERT FIGURE 4 HERE

PLEASE INSERT FIGURE 5 HERE

PLEASE INSERT FIGURE 6 HERE

PLEASE INSERT FIGURE 7 HERE

##  7 Illustrative example

In this section, we present an illustrative example of the implementation of
the RR${}_{r,s}-\gamma^{2}$ control charts in the presence of the measurement
error. The real industrial data from a sintering process in an Italian company
that manufactures sintered mechanical parts, which were introduced in [5], are
considered.

The process manufactures parts guarantee a pressure test by dropping time
$T_{pd}$ from 2 bar to 1.5 bar larger than 30 sec as a quality characteristic
related to the pore shrinkage. Since the presence of a constant
proportionality $\sigma_{pd}=\gamma_{pd}\times\mu_{pd}$ between the standard
deviation of the pressure drop time and its mean had been demonstrated by the
preliminary regression study relating $T_{pd}$ to the quantity $Q_{C}$ of
molten copper, the quality practitioners decide to monitor the CV
$\gamma_{pd}=\sigma_{pd}/\mu_{pd}$ to detect changes in the process
variability. According to the description in [5], an estimate
$\hat{\gamma}_{0}=0.417$ is calculated from a Phase I dataset based on a root
mean square computation. The phase II data are reproduced in Table 8.

According to [12] under the presence of the measurement error, we suppose that
the parameters of the linear covariate error model are $\eta=0.28$,
$\theta=0.05$, $B=1$, and $m=1$. Based on the process engineer’s experience, a
specific shift size $\tau=1.25$ was expected to detect from the process.
Therefore, we apply the upper-sided RR${}_{r,s-\gamma^{2}}$ control chart to
monitor the CV squared. The control limits of the RR${}^{+}_{2,3}-\gamma^{2}$,
RR${}^{+}_{3,4}-\gamma^{2}$ and RR${}_{4,5}-\gamma^{2}$ chart are found to be
$UCL^{+}=0.5567$, $UCL^{+}=0.3821$ and $UCL^{+}=0.2972$, respectively. The
values of $\gamma^{*2}_{i}$ are then plotted in these charts (Figure 8) long
with the control limit $UCL^{+}$. For purpose of comparison, we also draw the
control limit ($UCL^{+}=1.1913$)of the original Shewhart control chart with
the same parameters.

As can be seen from the Figure 8, the RR${}^{+}_{2,3}-\gamma^{2}$,
RR${}^{+}_{3,4}-\gamma^{2}$ and RR${}_{4,5}-\gamma^{2}$ chart signal the
occurrence of the out-of-control condition by two-out-of-three, three-out-of-
four, and four-out-of-five (respectively) successive plotting points above the
corresponding control limits from the sample #12. Meanwhile, the Shewhart
chart fails to detect this out-of-control condition.

PLEASE INSERT TABLE 8 HERE

PLEASE INSERT FIGURE 8 HERE

##  8 Concluding remarks

In this paper, the performance of Run Rules control charts is improved
slightly by monitoring the CV squared with the two one-sided charts rather
than monitoring directly the CV with a two-sided chart as in a previous study
in the literature. The effect of measurement errors on the performance of the
RR${}_{r,s}-{\gamma}^{2}$ control charts using a linear covariate error model
is also investigated. We have pointed out the negative effect of measurement
errors on the proposed charts: the increase of $\eta$ and $\theta$ leads to
the increase of $EARL$. Moreover, the obtained results show that measuring
repeatedly is not an efficient method for limiting these unexpected effects
HANH HEO.

## References

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Figure 1: The approximate density function of the $\hat{\gamma}^{2}$ for
$n=5$. Figure 2: The effect of $\theta$ and $\eta$ on the performance of the
RR${}_{r,s}-\gamma^{2}$ control charts in the presence of measurement errors
for $\gamma_{0}=0.05$.  Figure 3: The effect of $\theta$ and $\eta$ on the
performance of the RR${}_{r,s}-\gamma^{2}$ control charts in the presence of
measurement error for $\gamma_{0}=0.2$. Figure 4: The effect of $B$ on the
performance of the RR${}_{r,s}-\gamma^{2}$ control charts in the presence of
measurement errors for $\gamma_{0}=0.05$; $n=5$ (-$\square$-) and $n=15$
($-\blacksquare-$).  Figure 5: The effect of $B$ on the performance of the
RR${}_{r,s}-\gamma^{2}$ control charts in the presence of measurement error
for $\gamma_{0}=0.2$; $n=5$ (-$\square$-) and $n=15$ ($-\blacksquare-$).
Figure 6: The effect of $m$ on the performance of the RR${}_{r,s}-\gamma^{2}$
control charts in the presence of measurement errors for $\gamma_{0}=0.05$;
$n=5$ (-$\square$-) and $n=15$ ($-\blacksquare-$).  Figure 7: The effect of
$m$ on the performance of the RR${}_{r,s}-\gamma^{2}$ control charts in the
presence of measurement error for $\gamma_{0}=0.2$; $n=5$ (-$\square$-) and
$n=15$ ($-\blacksquare-$).

Figure 8: The upward CUSUM-$\gamma^{2}$ control chart in the presence of the
measurement error corresponding to the Phase II data in Table 8.

$\eta$ | $\theta$ | $\gamma_{0}$ | $n=10$ | $n=15$ | $n=10$ | $n=15$ | $n=10$ | $n=15$
---|---|---|---|---|---|---|---|---
|  |  | RR${}_{2,3}-\gamma^{2}$ | RR${}_{3,4}-\gamma^{2}$ | RR${}_{3,4}-\gamma^{2}$
0.1 | $0.01$ | $0.05$ | $0.0004$ | $0.0011$ | $0.0007$ | $0.0014$ | $0.0009$ | $0.0016$
|  |  | $0.0063$ | $0.0044$ | $0.0047$ | $0.0037$ | $0.0039$ | $0.0033$
|  | $0.10$ | $0.0015$ | $0.0043$ | $0.0027$ | $0.0055$ | $0.0038$ | $0.0065$
|  |  | $0.0254$ | $0.0175$ | $0.0191$ | $0.0148$ | $0.0156$ | $0.0132$
|  |  | $0.0582$ | $0.0399$ | $0.0435$ | $0.0336$ | $0.0354$ | $0.0299$
|  | $0.20$ | $0.0059$ | $0.0172$ | $0.0107$ | $0.0220$ | $0.0151$ | $0.0257$
|  |  | $0.1061$ | $0.0718$ | $0.0786$ | $0.0602$ | $0.0636$ | $0.0534$
$0.28$ | $0.05$ | $0.05$ | $0.0004$ | $0.0011$ | $0.0007$ | $0.0014$ | $0.0009$ | $0.0016$
|  |  | $0.0062$ | $0.0043$ | $0.0047$ | $0.0036$ | $0.0038$ | $0.0033$
|  | $0.10$ | $0.0015$ | $0.0043$ | $0.0027$ | $0.0055$ | $0.0037$ | $0.0064$
|  |  | $0.0251$ | $0.0173$ | $0.0189$ | $0.0146$ | $0.0154$ | $0.0130$
|  |  | $0.0575$ | $0.0394$ | $0.0430$ | $0.0332$ | $0.0350$ | $0.0295$
|  | $0.20$ | $0.0059$ | $0.0170$ | $0.0106$ | $0.0218$ | $0.0149$ | $0.0254$
|  |  | $0.1047$ | $0.0709$ | $0.0776$ | $0.0595$ | $0.0629$ | $0.0528$

Table 1: Values of $LCL$ (first row) and $UCL$ (second row) for the
RR${}_{r,s}-\gamma^{2}$ control charts in the presence of measurement errors,
for different values of $\eta$, $\theta$, $n$, $\gamma_{0}$, $B=1$ and $m=1$.

Charts | $\gamma_{0}=0.05$ | $\gamma_{0}=0.1$ | $\gamma_{0}=0.2$
---|---|---|---
$n=5$ | $n=15$ | $n=5$ | $n=15$ | $n=5$ | $n=15$
RR${}_{2,3}-{\gamma}^{2}$ | $(1.194,2.167)$ | $(1.487,2.010)$ | $(1.170,2.183)$ | $(1.464,2.023)$ | $(1.088,2.245)$ | $(1.407,2.069)$
RR${}_{3,4}-{\gamma}^{2}$ | $(1.023,1.293)$ | $(1.159,1.298)$ | $(1.003,1.301)$ | $(1.143,1.306)$ | $(0.930,1.331)$ | $(1.098,1.333)$
RR${}_{4,5}-{\gamma}^{2}$ | $(0.866,0.801)$ | $(0.915,0.872)$ | $(0.849,0.808)$ | $(0.902,0.878)$ | $(0.785,0.832)$ | $(0.865,0.899)$

Table 2: Values of the parameters $k_{d}$ (left side) and $k_{u}$ (right size)
of the downward chart and the upward RR${}_{r,s}-{\gamma}^{2}$ charts for
$\gamma_{0}=\\{0.05,0.1,0.2\\}$ and $n=\\{5,15\\}$ when $ARL_{0}=370.4$.

Charts | $\tau$ | $\gamma_{0}=0.05$ | $\gamma_{0}=0.1$ | $\gamma_{0}=0.2$
---|---|---|---|---
$n=5$ | $n=15$ | $n=5$ | $n=15$ | $n=5$ | $n=15$
RR${}_{2,3}^{-}-{\gamma}^{2}$ | 0.5 | $(8.1,6.6)$ | $(2.1,0.3)$ | $(8.1,6.5)$ | $(2.1,0.3)$ | $(8.2,6.7)$ | $(2.1,0.3)$
RR${}_{3,4}^{-}-{\gamma}^{2}$ | $(5.6,3.3)$ | $(3.0,0.1)$ | $(5.6,3.3)$ | $(3.0,0.1)$ | $(5.7,3.4)$ | $(3.0,0.1)$
RR${}_{4,5}^{-}-{\gamma}^{2}$ | $(5.4,2.1)$ | $(4.0,0.1)$ | $(5.4,2.1)$ | $(4.0,0.1)$ | $(5.4,2.2)$ | $(4.0,0.1)$
RR${}_{2,3}^{-}-{\gamma}^{2}$ | 0.65 | $(26.9,25.2)$ | $(3.8,2.3)$ | $(26.6,25.0)$ | $(3.8,2.2)$ | $(27.1,25.5)$ | $(3.9,2.4)$
RR${}_{3,4}^{-}-{\gamma}^{2}$ | $((16.2,13.7)$ | $(3.8,1.4)$ | $(16.2,13.8)$ | $(3.8,1.4)$ | $(16.6,14.1)$ | $(3.9,1.5)$
RR${}_{4,5}^{-}-{\gamma}^{2}$ | $(12.6,9.5)$ | $(4.5,1.0)$ | $(12.7,9.6)$ | $(4.5,1.0)$ | $(13.0,9.9)$ | $(4.5,1.0)$
RR${}_{2,3}^{-}-{\gamma}^{2}$ | 0.8 | $(87.9,86.1)$ | $(17.9,16.2)$ | $(87.2,85.4)$ | $(17.5,15.9)$ | $(88.4,86.6)$ | $(18.2,16.5)$
RR${}_{3,4}^{-}-{\gamma}^{2}$ | $(59.8,57.1)$ | $(12.6,10.2)$ | $(59.9,57.2)$ | $(12.7,10.3)$ | $(61.1,58.4)$ | $(13.2,10.8)$
RR${}_{4,5}^{-}-{\gamma}^{2}$ | $(46.8,43.4)$ | $(11.1,8.0)$ | $(47.0,43.6)$ | $(11.2,8.0)$ | $(48.1,44.7)$ | $(11.6,8.4)$
RR${}_{2,3}^{-}-{\gamma}^{2}$ | 0.9 | $(184.4,182.5)$ | $(75.4,73.7)$ | $(183.7,181.9)$ | $(73.9,72.2)$ | $(185.2,183.4)$ | $(75.7,73.9)$
RR${}_{3,4}^{-}-{\gamma}^{2}$ | $(149.3,146.5)$ | $(55.4,52.7)$ | $(149.5,146.7)$ | $(55.3,52.6)$ | $(151.3,148.5)$ | $(57.1,54.5)$
RR${}_{4,5}^{-}-{\gamma}^{2}$ | $(128.7,125.1)$ | $(46.5,43.1)$ | $(129.1,125.4)$ | $(46.7,43.3)$ | $(130.9,127.3)$ | $(48.3,44.9)$
RR${}_{2,3}^{+}-{\gamma}^{2}$ | 1.10 | $(95.9,94.1)$ | $(45.8,44.1)$ | $(96.5,94.7)$ | $(46.4,44.6)$ | $(98.7,96.9)$ | $(48.5,46.8)$
RR${}_{3,4}^{+}-{\gamma}^{2}$ | $(94.2,91.5)$ | $(42.3,39.7)$ | $(94.8,92.0)$ | $(42.8,40.1)$ | $(97.0,94.2)$ | $(44.7,42.1)$
RR${}_{4,5}^{+}-{\gamma}^{2}$ | $(94.9,91.4)$ | $(41.3,37.9)$ | $(95.4,91.9)$ | $(41.7,38.3)$ | $(97.6,94.0)$ | $(43.6,40.2)$
RR${}_{2,3}^{+}-{\gamma}^{2}$ | 1.25 | $(25.8,24.2)$ | $(8.7,7.1)$ | $(26.1,24.4)$ | $(8.8,7.2)$ | $(27.2,25.6)$ | $(9.4,7.8)$
RR${}_{3,4}^{+}-{\gamma}^{2}$ | $(26.3,23.8)$ | $(8.9,6.5)$ | $(26.5,24.0)$ | $(9.0,6.6)$ | $(27.6,25.1)$ | $(9.5,7.1)$
RR${}_{4,5}^{+}-{\gamma}^{2}$ | $(27.5,24.2)$ | $(9.5,6.4)$ | $(27.8,24.5)$ | $(9.6,6.5)$ | $(28.9,25.5)$ | $(10.1,7.0)$
RR${}_{2,3}^{+}-{\gamma}^{2}$ | 1.5 | $(8.1,6.6)$ | $(3.1,1.5)$ | $(8.2,6.7)$ | $(3.1,1.6)$ | $(8.6,7.1)$ | $(3.3,1.7)$
RR${}_{3,4}^{+}-{\gamma}^{2}$ | $(9.1,6.7)$ | $(3.9,1.4)$ | $(9.2,6.8)$ | $(3.9,1.5)$ | $(9.6,7.2)$ | $(4.0,1.6)$
RR${}_{4,5}^{+}-{\gamma}^{2}$ | $(10.2,7.1)$ | $(4.8,1.4)$ | $(10.3,7.2)$ | $(4.8,1.4)$ | $(10.7,7.6)$ | $(4.9,1.6)$
RR${}_{2,3}^{+}-{\gamma}^{2}$ | 2.0 | $(3.4,1.9)$ | $(2.1,0.3)$ | $(3.4,1.9)$ | $(2.1,0.3)$ | $(3.6,2.1)$ | $(2.1,0.4)$
RR${}_{3,4}^{+}-{\gamma}^{2}$ | $(4.3,2.0)$ | $(3.1,0.3)$ | $(4.4,2.0)$ | $(3.1,0.3)$ | $(4.6,2.2)$ | $(3.1,0.3)$
RR${}_{4,5}^{+}-{\gamma}^{2}$ | $(5.3,2.1)$ | $(4.0,0.2)$ | $(5.4,2.1)$ | $(4.0,0.2)$ | $(5.6,2.3)$ | $(4.1,0.3)$

Table 3: Values of $(ARL_{1},SDRL_{1})$ of RR${}_{r,s}-{\gamma}^{2}$ charts
corresponding to the chart parameters in Table 1 for various situations of
$\tau$.

Charts | $\tau$ | $\eta=0$ | $\eta=0.1$ | $\eta=0.2$ | $\eta=0.3$ | $\eta=0.5$ | $\eta=1$
---|---|---|---|---|---|---|---
| $n=5$ |  |
| $n=5$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(8.92,8.85,8.98)$ | $(8.94,8.85,8.99)$ | $(8.92,8.84,8.99)$ | $(8.92,8.86,9.01)$ | $(8.90,8.84,9.06)$ | $(8.87,8.87,9.24)$
$0.7$ | $(29.38,29.11,29.54)$ | $(29.45,29.14,29.58)$ | $(29.35,29.06,29.57)$ | $(29.40,29.13,29.65)$ | $(29.29,29.09,29.80)$ | $(29.19,29.18,30.42)$
$0.8$ | $(93.12,92.52,93.41)$ | $(93.28,92.44,93.58)$ | $(93.15,92.24,93.38)$ | $(93.20,92.54,93.68)$ | $(92.90,92.39,93.98)$ | $(92.74,92.60,95.22)$
$1.3$ | $(28.57,28.83,29.93)$ | $(28.57,28.84,29.94)$ | $(28.57,28.85,29.99)$ | $(28.58,28.87,30.06)$ | $(28.59,28.92,30.30)$ | $(28.66,29.19,31.45)$
$1.5$ | $(9.07,9.17,9.62)$ | $(9.07,9.17,9.62)$ | $(9.07,9.18,9.64)$ | $(9.07,9.19,9.67)$ | $(9.07,9.21,9.77)$ | $(9.10,9.32,10.24)$
$2.0$ | $(3.70,3.74,3.92)$ | $(3.70,3.74,3.92)$ | $(3.70,3.74,3.92)$ | $(3.70,3.75,3.94)$ | $(3.70,3.76,3.98)$ | $(3.71,3.80,4.16)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(6.01,6.02,6.11)$ | $(6.02,6.02,6.12)$ | $(6.02,6.02,6.12)$ | $(6.01,6.02,6.13)$ | $(6.01,6.02,6.15)$ | $(6.01,6.05,6.26)$
$0.7$ | $(17.69,17.71,18.09)$ | $(17.71,17.71,18.11)$ | $(17.71,17.71,18.11)$ | $(17.70,17.73,18.15)$ | $(17.71,17.73,18.25)$ | $(17.68,17.82,18.68)$
$0.8$ | $(64.10,64.21,65.33)$ | $(64.20,64.19,65.35)$ | $(64.22,64.17,65.37)$ | $(64.13,64.24,65.47)$ | $(64.22,64.25,65.78)$ | $(64.07,64.48,67.06)$
$1.3$ | $(28.92,29.17,30.21)$ | $(28.92,29.17,30.22)$ | $(28.92,29.18,30.26)$ | $(28.93,29.20,30.34)$ | $(28.93,29.25,30.56)$ | $(29.00,29.51,31.63)$
$1.5$ | $(10.01,10.12,10.55)$ | $(10.01,10.12,10.55)$ | $(10.01,10.12,10.57)$ | $(10.02,10.13,10.60)$ | $(10.02,10.15,10.69)$ | $(10.05,10.26,11.14)$
$2.0$ | $(4.67,4.71,4.89)$ | $(4.67,4.71,4.89)$ | $(4.67,4.72,4.90)$ | $(4.67,4.72,4.91)$ | $(4.68,4.73,4.95)$ | $(4.69,4.77,5.13)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(5.64,5.65,5.71)$ | $(5.64,5.65,5.71)$ | $(5.64,5.65,5.72)$ | $(5.64,5.65,5.72)$ | $(5.64,5.65,5.74)$ | $(5.64,5.67,5.81)$
$0.7$ | $(13.75,13.79,14.09)$ | $(13.76,13.80,14.10)$ | $(13.76,13.81,14.12)$ | $(13.75,13.81,14.13)$ | $(13.76,13.81,14.20)$ | $(13.76,13.89,14.52)$
$0.8$ | $(50.44,50.60,51.63)$ | $(50.49,50.63,51.67)$ | $(50.51,50.66,51.74)$ | $(50.45,50.66,51.78)$ | $(50.46,50.67,52.03)$ | $(50.48,50.96,53.14)$
$1.3$ | $(30.17,30.42,31.45)$ | $(30.18,30.43,31.46)$ | $(30.19,30.43,31.51)$ | $(30.18,30.45,31.57)$ | $(30.20,30.51,31.80)$ | $(30.25,30.76,32.85)$
$1.5$ | $(11.17,11.27,11.70)$ | $(11.17,11.27,11.71)$ | $(11.17,11.27,11.73)$ | $(11.17,11.28,11.76)$ | $(11.17,11.30,11.85)$ | $(11.20,11.41,12.30)$
$2.0$ | $(5.68,5.72,5.90)$ | $(5.68,5.72,5.90)$ | $(5.68,5.72,5.91)$ | $(5.68,5.72,5.92)$ | $(5.68,5.73,5.96)$ | $(5.69,5.78,6.14)$
| $n=15$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.14)$ | $(2.12,2.12,2.15)$
$0.7$ | $(4.12,4.08,4.18)$ | $(4.12,4.08,4.18)$ | $(4.12,4.08,4.18)$ | $(4.11,4.08,4.19)$ | $(4.11,4.08,4.22)$ | $(4.09,4.10,4.35)$
$0.8$ | $(19.86,19.42,19.97)$ | $(19.83,19.43,20.02)$ | $(19.83,19.45,20.02)$ | $(19.80,19.45,20.06)$ | $(19.73,19.43,20.20)$ | $(19.56,19.55,21.01)$
$1.3$ | $(9.68,9.81,10.41)$ | $(9.68,9.82,10.41)$ | $(9.68,9.82,10.44)$ | $(9.68,9.83,10.48)$ | $(9.69,9.86,10.61)$ | $(9.72,10.01,11.22)$
$1.5$ | $(3.32,3.36,3.52)$ | $(3.32,3.36,3.52)$ | $(3.32,3.36,3.53)$ | $(3.32,3.36,3.54)$ | $(3.32,3.37,3.58)$ | $(3.33,3.41,3.75)$
$2.0$ | $(2.13,2.14,2.17)$ | $(2.13,2.14,2.17)$ | $(2.13,2.14,2.18)$ | $(2.13,2.14,2.18)$ | $(2.13,2.14,2.19)$ | $(2.13,2.15,2.23)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.03,3.03)$
$0.7$ | $(4.02,4.02,4.09)$ | $(4.02,4.02,4.09)$ | $(4.01,4.02,4.10)$ | $(4.01,4.02,4.10)$ | $(4.01,4.03,4.12)$ | $(4.01,4.04,4.20)$
$0.8$ | $(13.91,13.92,14.41)$ | $(13.92,13.91,14.42)$ | $(13.90,13.91,14.44)$ | $(13.90,13.92,14.48)$ | $(13.89,13.95,14.59)$ | $(13.87,14.06,15.13)$
$1.3$ | $(9.77,9.89,10.42)$ | $(9.77,9.90,10.42)$ | $(9.78,9.90,10.44)$ | $(9.78,9.91,10.48)$ | $(9.78,9.94,10.59)$ | $(9.81,10.07,11.13)$
$1.5$ | $(4.10,4.13,4.28)$ | $(4.10,4.13,4.28)$ | $(4.10,4.14,4.29)$ | $(4.10,4.14,4.30)$ | $(4.10,4.14,4.33)$ | $(4.11,4.18,4.48)$
$2.0$ | $(3.09,3.10,3.12)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.14)$ | $(3.09,3.11,3.17)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$
$0.7$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.65)$ | $(4.58,4.59,4.66)$ | $(4.58,4.60,4.71)$
$0.8$ | $(12.09,12.14,12.55)$ | $(12.09,12.14,12.56)$ | $(12.09,12.14,12.58)$ | $(12.09,12.15,12.61)$ | $(12.09,12.17,12.70)$ | $(12.09,12.27,13.15)$
$1.3$ | $(10.37,10.48,10.97)$ | $(10.37,10.48,10.98)$ | $(10.37,10.49,11.00)$ | $(10.37,10.49,11.03)$ | $(10.38,10.52,11.14)$ | $(10.40,10.64,11.64)$
$1.5$ | $(4.97,5.00,5.13)$ | $(4.97,5.00,5.13)$ | $(4.97,5.00,5.14)$ | $(4.97,5.00,5.15)$ | $(4.97,5.01,5.18)$ | $(4.98,5.04,5.32)$
$2.0$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.11)$ | $(4.07,4.08,4.13)$

Table 4: The $ARL$ values of the RR${}_{r,s}-\gamma^{2}$ control charts in the
presence of measurement errors for $\gamma_{0}=0.05$ (left side),
$\gamma_{0}=0.1$ (middle) and $\gamma_{0}=0.2$ (right side), and for different
values of $\eta$, $\theta=0.05$, $\tau$, $n$, $B=1$, $m=1$.

Charts | $\tau$ | $\theta=0$ | $\theta=0.01$ | $\theta=0.02$ | $\theta=0.03$ | $\theta=0.04$ | $\theta=0.05$
---|---|---|---|---|---|---|---
| $n=5$ |  |
| $n=5$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(8.11,8.06,8.22)$ | $(8.28,8.21,8.38)$ | $(8.43,8.35,8.53)$ | $(8.59,8.52,8.69)$ | $(8.75,8.69,8.84)$ | $(8.93,8.85,9.01)$
$0.7$ | $(26.88,26.65,27.19)$ | $(27.38,27.15,27.71)$ | $(27.87,27.55,28.15)$ | $(28.36,28.10,28.66)$ | $(28.87,28.62,29.12)$ | $(29.42,29.13,29.64)$
$0.8$ | $(87.72,87.23,88.36)$ | $(88.96,88.30,89.56)$ | $(89.94,89.12,90.51)$ | $(90.85,90.33,91.69)$ | $(91.95,91.30,92.44)$ | $(93.24,92.53,93.63)$
$1.3$ | $(25.84,26.13,27.35)$ | $(26.38,26.67,27.88)$ | $(26.93,27.21,28.42)$ | $(27.47,27.76,28.96)$ | $(28.02,28.31,29.50)$ | $(28.58,28.86,30.04)$
$1.5$ | $(8.09,8.20,8.68)$ | $(8.28,8.39,8.87)$ | $(8.47,8.59,9.07)$ | $(8.67,8.78,9.26)$ | $(8.87,8.98,9.46)$ | $(9.07,9.18,9.66)$
$2.0$ | $(3.38,3.42,3.61)$ | $(3.44,3.48,3.67)$ | $(3.50,3.55,3.73)$ | $(3.57,3.61,3.80)$ | $(3.63,3.68,3.87)$ | $(3.70,3.75,3.93)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(5.60,5.61,5.71)$ | $(5.68,5.69,5.80)$ | $(5.76,5.77,5.88)$ | $(5.84,5.85,5.96)$ | $(5.93,5.93,6.04)$ | $(6.01,6.02,6.12)$
$0.7$ | $(16.15,16.20,16.62)$ | $(16.47,16.50,16.93)$ | $(16.76,16.78,17.24)$ | $(17.07,17.10,17.53)$ | $(17.38,17.40,17.84)$ | $(17.70,17.72,18.14)$
$0.8$ | $(59.75,59.92,61.22)$ | $(60.70,60.79,62.12)$ | $(61.56,61.53,63.02)$ | $(62.38,62.49,63.79)$ | $(63.26,63.35,64.67)$ | $(64.20,64.24,65.45)$
$1.3$ | $(26.28,26.56,27.72)$ | $(26.80,27.08,28.23)$ | $(27.33,27.60,28.75)$ | $(27.86,28.13,29.27)$ | $(28.39,28.66,29.79)$ | $(28.92,29.19,30.31)$
$1.5$ | $(9.05,9.16,9.63)$ | $(9.24,9.35,9.82)$ | $(9.43,9.54,10.01)$ | $(9.62,9.73,10.20)$ | $(9.82,9.93,10.39)$ | $(10.02,10.13,10.59)$
$2.0$ | $(4.35,4.39,4.58)$ | $(4.41,4.46,4.64)$ | $(4.48,4.52,4.70)$ | $(4.54,4.58,4.77)$ | $(4.61,4.65,4.84)$ | $(4.67,4.72,4.91)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(5.37,5.39,5.46)$ | $(5.43,5.44,5.51)$ | $(5.48,5.49,5.56)$ | $(5.53,5.54,5.61)$ | $(5.58,5.59,5.67)$ | $(5.64,5.65,5.72)$
$0.7$ | $(12.65,12.70,13.04)$ | $(12.87,12.92,13.25)$ | $(13.09,13.14,13.47)$ | $(13.31,13.35,13.68)$ | $(13.53,13.58,13.90)$ | $(13.76,13.80,14.13)$
$0.8$ | $(46.80,47.00,48.23)$ | $(47.56,47.75,48.90)$ | $(48.26,48.47,49.65)$ | $(49.00,49.14,50.35)$ | $(49.70,49.91,51.07)$ | $(50.47,50.63,51.74)$
$1.3$ | $(27.55,27.82,28.97)$ | $(28.07,28.34,29.48)$ | $(28.60,28.86,29.99)$ | $(29.13,29.39,30.51)$ | $(29.65,29.92,31.04)$ | $(30.19,30.44,31.56)$
$1.5$ | $(10.18,10.30,10.77)$ | $(10.38,10.49,10.96)$ | $(10.57,10.68,11.15)$ | $(10.77,10.88,11.35)$ | $(10.97,11.08,11.55)$ | $(11.17,11.28,11.75)$
$2.0$ | $(5.34,5.39,5.58)$ | $(5.41,5.45,5.64)$ | $(5.47,5.52,5.71)$ | $(5.54,5.58,5.77)$ | $(5.61,5.65,5.84)$ | $(5.68,5.72,5.91)$
| $n=15$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(2.09,2.09,2.10)$ | $(2.09,2.09,2.10)$ | $(2.10,2.10,2.11)$ | $(2.11,2.11,2.12)$ | $(2.11,2.11,2.13)$ | $(2.12,2.12,2.13)$
$0.7$ | $(3.79,3.76,3.87)$ | $(3.85,3.82,3.94)$ | $(3.92,3.89,4.00)$ | $(3.98,3.95,4.06)$ | $(4.04,4.02,4.12)$ | $(4.12,4.08,4.19)$
$0.8$ | $(17.85,17.59,18.24)$ | $(18.22,17.94,18.60)$ | $(18.65,18.32,18.96)$ | $(19.00,18.68,19.31)$ | $(19.39,19.08,19.68)$ | $(19.82,19.40,20.05)$
$1.3$ | $(8.66,8.81,9.44)$ | $(8.86,9.01,9.64)$ | $(9.06,9.21,9.84)$ | $(9.26,9.41,10.05)$ | $(9.47,9.62,10.26)$ | $(9.68,9.83,10.47)$
$1.5$ | $(3.07,3.11,3.28)$ | $(3.12,3.16,3.33)$ | $(3.17,3.21,3.38)$ | $(3.22,3.26,3.43)$ | $(3.27,3.31,3.48)$ | $(3.32,3.36,3.54)$
$2.0$ | $(2.08,2.09,2.13)$ | $(2.09,2.10,2.14)$ | $(2.10,2.11,2.15)$ | $(2.11,2.12,2.16)$ | $(2.12,2.13,2.17)$ | $(2.13,2.14,2.18)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(3.01,3.02,3.02)$ | $(3.02,3.02,3.02)$ | $(3.02,3.02,3.02)$ | $(3.02,3.02,3.02)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$
$0.7$ | $(3.83,3.84,3.91)$ | $(3.87,3.87,3.95)$ | $(3.90,3.91,3.99)$ | $(3.94,3.95,4.02)$ | $(3.98,3.98,4.06)$ | $(4.02,4.02,4.10)$
$0.8$ | $(12.64,12.68,13.22)$ | $(12.89,12.92,13.47)$ | $(13.14,13.16,13.71)$ | $(13.38,13.41,13.97)$ | $(13.64,13.67,14.22)$ | $(13.91,13.92,14.46)$
$1.3$ | $(8.87,9.00,9.56)$ | $(9.04,9.18,9.74)$ | $(9.22,9.36,9.92)$ | $(9.41,9.54,10.10)$ | $(9.59,9.72,10.28)$ | $(9.78,9.91,10.47)$
$1.5$ | $(3.88,3.91,4.06)$ | $(3.92,3.96,4.11)$ | $(3.97,4.00,4.15)$ | $(4.01,4.04,4.20)$ | $(4.06,4.09,4.25)$ | $(4.10,4.14,4.29)$
$2.0$ | $(3.06,3.06,3.09)$ | $(3.07,3.07,3.10)$ | $(3.07,3.08,3.10)$ | $(3.08,3.08,3.11)$ | $(3.09,3.09,3.12)$ | $(3.09,3.10,3.13)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(4.00,4.00,4.00)$ | $(4.00,4.00,4.01)$ | $(4.00,4.00,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$
$0.7$ | $(4.46,4.47,4.52)$ | $(4.48,4.49,4.54)$ | $(4.51,4.51,4.57)$ | $(4.53,4.54,4.59)$ | $(4.55,4.56,4.62)$ | $(4.58,4.59,4.64)$
$0.8$ | $(11.10,11.17,11.62)$ | $(11.29,11.36,11.82)$ | $(11.48,11.55,12.01)$ | $(11.69,11.75,12.21)$ | $(11.89,11.94,12.40)$ | $(12.09,12.14,12.60)$
$1.3$ | $(9.50,9.63,10.15)$ | $(9.67,9.79,10.32)$ | $(9.84,9.97,10.49)$ | $(10.02,10.14,10.67)$ | $(10.19,10.31,10.85)$ | $(10.37,10.49,11.02)$
$1.5$ | $(4.77,4.80,4.93)$ | $(4.81,4.84,4.97)$ | $(4.85,4.88,5.02)$ | $(4.89,4.92,5.06)$ | $(4.93,4.96,5.10)$ | $(4.97,5.00,5.15)$
$2.0$ | $(4.05,4.05,4.07)$ | $(4.05,4.05,4.07)$ | $(4.06,4.06,4.08)$ | $(4.06,4.06,4.09)$ | $(4.07,4.07,4.09)$ | $(4.07,4.08,4.10)$

Table 5: The $ARL$ values of the RR${}_{r,s}-\gamma^{2}$ control charts in the
presence of measurement errors for $\gamma_{0}=0.05$ (left side),
$\gamma_{0}=0.1$ (middle) and $\gamma_{0}=0.2$ (right side), and for different
values of $\theta$, $\eta=0.28$, $\tau$, $n$, $B=1$, $m=1$.

| $n=5$ |
---|---|---
Charts | $\tau$ | $B=0.8$ | $B=0.9$ | $B=1.0$ | $B=1.1$ | $B=1.2$
| $n=5$ |
RR${}_{2,3}-\gamma^{2}$ | 0.5 | $(9.13,9.05,9.22)$ | $(9.03,8.93,9.11)$ | $(8.93,8.85,9.01)$ | $(8.84,8.76,8.94)$ | $(8.79,8.72,8.87)$
0.7 | $(30.00,29.74,30.26)$ | $(29.73,29.33,29.95)$ | $(29.42,29.13,29.64)$ | $(29.10,28.82,29.42)$ | $(28.99,28.71,29.24)$
0.8 | $(94.27,93.68,94.95)$ | $(93.89,92.84,94.31)$ | $(93.24,92.53,93.63)$ | $(92.45,91.72,93.28)$ | $(92.23,91.63,92.81)$
1.3 | $(29.28,29.57,30.79)$ | $(28.89,29.17,30.37)$ | $(28.58,28.86,30.04)$ | $(28.33,28.60,29.77)$ | $(28.11,28.39,29.55)$
1.5 | $(9.33,9.44,9.95)$ | $(9.18,9.30,9.79)$ | $(9.07,9.18,9.66)$ | $(8.98,9.09,9.56)$ | $(8.90,9.01,9.48)$
2.0 | $(3.79,3.83,4.03)$ | $(3.74,3.78,3.98)$ | $(3.70,3.75,3.93)$ | $(3.67,3.71,3.90)$ | $(3.64,3.69,3.87)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(6.12,6.13,6.24)$ | $(6.06,6.06,6.18)$ | $(6.01,6.02,6.12)$ | $(5.98,5.98,6.09)$ | $(5.94,5.95,6.05)$
$0.7$ | $(18.08,18.11,18.55)$ | $(17.89,17.88,18.33)$ | $(17.70,17.72,18.14)$ | $(17.57,17.58,18.00)$ | $(17.44,17.46,17.88)$
$0.8$ | $(65.22,65.28,66.56)$ | $(64.70,64.64,66.01)$ | $(64.20,64.24,65.45)$ | $(63.85,63.84,65.08)$ | $(63.43,63.53,64.76)$
$1.3$ | $(29.60,29.88,31.04)$ | $(29.22,29.50,30.63)$ | $(28.92,29.19,30.31)$ | $(28.68,28.94,30.06)$ | $(28.48,28.74,29.84)$
$1.5$ | $(10.27,10.38,10.87)$ | $(10.13,10.24,10.71)$ | $(10.02,10.13,10.59)$ | $(9.93,10.03,10.49)$ | $(9.85,9.96,10.41)$
$2.0$ | $(4.76,4.81,5.00)$ | $(4.71,4.76,4.95)$ | $(4.67,4.72,4.91)$ | $(4.64,4.69,4.87)$ | $(4.62,4.66,4.84)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(5.71,5.72,5.79)$ | $(5.67,5.68,5.75)$ | $(5.64,5.65,5.72)$ | $(5.61,5.62,5.69)$ | $(5.59,5.60,5.67)$
$0.7$ | $(14.04,14.08,14.43)$ | $(13.88,13.93,14.26)$ | $(13.76,13.80,14.13)$ | $(13.66,13.70,14.02)$ | $(13.57,13.61,13.93)$
$0.8$ | $(51.38,51.54,52.76)$ | $(50.91,51.07,52.19)$ | $(50.47,50.63,51.74)$ | $(50.18,50.32,51.43)$ | $(49.86,50.02,51.15)$
$1.3$ | $(30.85,31.13,32.28)$ | $(30.48,30.75,31.88)$ | $(30.19,30.44,31.56)$ | $(29.94,30.20,31.30)$ | $(29.74,30.00,31.09)$
$1.5$ | $(11.42,11.54,12.03)$ | $(11.28,11.40,11.87)$ | $(11.17,11.28,11.75)$ | $(11.08,11.19,11.65)$ | $(11.00,11.11,11.57)$
$2.0$ | $(5.77,5.81,6.01)$ | $(5.72,5.76,5.96)$ | $(5.68,5.72,5.91)$ | $(5.65,5.69,5.88)$ | $(5.62,5.66,5.85)$
| $n=15$ |
RR${}_{2,3}-\gamma^{2}$ | 0.5 | $(2.13,2.13,2.14)$ | $(2.13,2.13,2.14)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.11,2.13)$
0.7 | $(4.20,4.17,4.29)$ | $(4.15,4.12,4.23)$ | $(4.12,4.08,4.19)$ | $(4.08,4.05,4.16)$ | $(4.06,4.02,4.13)$
0.8 | $(20.29,19.95,20.61)$ | $(20.05,19.69,20.26)$ | $(19.82,19.40,20.05)$ | $(19.64,19.25,19.86)$ | $(19.50,19.11,19.73)$
1.3 | $(9.95,10.11,10.77)$ | $(9.80,9.95,10.60)$ | $(9.68,9.83,10.47)$ | $(9.58,9.73,10.36)$ | $(9.50,9.65,10.28)$
1.5 | $(3.39,3.43,3.62)$ | $(3.35,3.39,3.57)$ | $(3.32,3.36,3.54)$ | $(3.30,3.34,3.51)$ | $(3.28,3.31,3.49)$
2.0 | $(2.14,2.15,2.19)$ | $(2.13,2.14,2.18)$ | $(2.13,2.14,2.18)$ | $(2.12,2.13,2.17)$ | $(2.12,2.13,2.17)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(3.03,3.03,3.03)$ | $(3.03,3.03,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$
$0.7$ | $(4.06,4.07,4.15)$ | $(4.04,4.04,4.12)$ | $(4.02,4.02,4.10)$ | $(4.00,4.00,4.08)$ | $(3.98,3.99,4.06)$
$0.8$ | $(14.22,14.25,14.82)$ | $(14.05,14.07,14.62)$ | $(13.91,13.92,14.46)$ | $(13.78,13.81,14.34)$ | $(13.68,13.71,14.23)$
$1.3$ | $(10.02,10.15,10.74)$ | $(9.88,10.02,10.59)$ | $(9.78,9.91,10.47)$ | $(9.69,9.82,10.38)$ | $(9.62,9.75,10.30)$
$1.5$ | $(4.16,4.20,4.36)$ | $(4.13,4.16,4.32)$ | $(4.10,4.14,4.29)$ | $(4.08,4.12,4.27)$ | $(4.06,4.10,4.25)$
$2.0$ | $(3.10,3.11,3.14)$ | $(3.10,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.09,3.12)$ | $(3.09,3.09,3.12)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$
$0.7$ | $(4.61,4.62,4.68)$ | $(4.59,4.60,4.66)$ | $(4.58,4.59,4.64)$ | $(4.57,4.58,4.63)$ | $(4.56,4.57,4.62)$
$0.8$ | $(12.34,12.40,12.88)$ | $(12.20,12.26,12.72)$ | $(12.09,12.14,12.60)$ | $(12.00,12.05,12.50)$ | $(11.92,11.98,12.42)$
$1.3$ | $(10.60,10.72,11.28)$ | $(10.47,10.60,11.14)$ | $(10.37,10.49,11.02)$ | $(10.29,10.41,10.93)$ | $(10.22,10.34,10.86)$
$1.5$ | $(5.03,5.06,5.21)$ | $(5.00,5.03,5.18)$ | $(4.97,5.00,5.15)$ | $(4.95,4.98,5.12)$ | $(4.94,4.97,5.11)$
$2.0$ | $(4.08,4.08,4.11)$ | $(4.08,4.08,4.11)$ | $(4.07,4.08,4.10)$ | $(4.07,4.07,4.10)$ | $(4.07,4.07,4.09)$

Table 6: The $ARL$ values of the RR${}_{r,s}-\gamma^{2}$ control charts in the
presence of measurement errors for $\gamma_{0}=0.05$ (left side),
$\gamma_{0}=0.1$ (middle) and $\gamma_{0}=0.2$ (right side), and for different
values of $B$, $\tau$, $n$, $\eta=0.28$, $\theta=0.05$, $m=1$.

| $n=5$ |
---|---|---
Charts | $\tau$ | $m=1$ | $m=3$ | $m=5$ | $m=7$ | $m=10$
| $n=5$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(8.93,8.85,9.01)$ | $(8.92,8.84,8.99)$ | $(8.93,8.84,8.99)$ | $(8.94,8.85,8.99)$ | $(8.93,8.84,8.99)$
$0.7$ | $(29.42,29.13,29.64)$ | $(29.39,29.06,29.56)$ | $(29.41,29.09,29.56)$ | $(29.42,29.11,29.58)$ | $(29.39,29.07,29.58)$
$0.8$ | $(93.24,92.53,93.63)$ | $(93.00,92.20,93.47)$ | $(93.22,92.44,93.43)$ | $(93.20,92.50,93.57)$ | $(93.12,92.23,93.52)$
$1.3$ | $(28.58,28.86,30.04)$ | $(28.57,28.84,29.96)$ | $(28.57,28.84,29.95)$ | $(28.57,28.83,29.95)$ | $(28.57,28.83,29.94)$
$1.5$ | $(9.07,9.18,9.66)$ | $(9.07,9.18,9.63)$ | $(9.07,9.17,9.62)$ | $(9.07,9.17,9.62)$ | $(9.07,9.17,9.62)$
$2.0$ | $(3.70,3.75,3.93)$ | $(3.70,3.74,3.92)$ | $(3.70,3.74,3.92)$ | $(3.70,3.74,3.92)$ | $(3.70,3.74,3.92)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(6.01,6.02,6.12)$ | $(6.02,6.02,6.12)$ | $(6.01,6.01,6.12)$ | $(6.02,6.02,6.12)$ | $(6.01,6.02,6.12)$
$0.7$ | $(17.70,17.72,18.14)$ | $(17.72,17.70,18.11)$ | $(17.70,17.69,18.11)$ | $(17.71,17.71,18.10)$ | $(17.69,17.71,18.11)$
$0.8$ | $(64.20,64.24,65.45)$ | $(64.26,64.15,65.33)$ | $(64.18,64.15,65.36)$ | $(64.18,64.21,65.35)$ | $(64.15,64.15,65.39)$
$1.3$ | $(28.92,29.19,30.31)$ | $(28.92,29.17,30.25)$ | $(28.92,29.17,30.23)$ | $(28.92,29.17,30.22)$ | $(28.92,29.17,30.22)$
$1.5$ | $(10.02,10.13,10.59)$ | $(10.01,10.12,10.56)$ | $(10.01,10.12,10.55)$ | $(10.01,10.12,10.55)$ | $(10.01,10.12,10.55)$
$2.0$ | $(4.67,4.72,4.91)$ | $(4.67,4.72,4.89)$ | $(4.67,4.71,4.89)$ | $(4.67,4.71,4.89)$ | $(4.67,4.71,4.89)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(5.64,5.65,5.72)$ | $(5.64,5.65,5.71)$ | $(5.64,5.65,5.71)$ | $(5.64,5.65,5.71)$ | $(5.64,5.65,5.71)$
$0.7$ | $(13.76,13.80,14.13)$ | $(13.76,13.80,14.10)$ | $(13.75,13.80,14.10)$ | $(13.76,13.79,14.10)$ | $(13.75,13.79,14.10)$
$0.8$ | $(50.47,50.63,51.74)$ | $(50.50,50.65,51.67)$ | $(50.46,50.62,51.68)$ | $(50.48,50.58,51.67)$ | $(50.47,50.60,51.67)$
$1.3$ | $(30.19,30.44,31.56)$ | $(30.18,30.43,31.49)$ | $(30.18,30.43,31.47)$ | $(30.18,30.42,31.46)$ | $(30.17,30.42,31.46)$
$1.5$ | $(11.17,11.28,11.75)$ | $(11.17,11.27,11.72)$ | $(11.17,11.27,11.71)$ | $(11.17,11.27,11.71)$ | $(11.17,11.27,11.71)$
$2.0$ | $(5.68,5.72,5.91)$ | $(5.68,5.72,5.90)$ | $(5.68,5.72,5.90)$ | $(5.68,5.72,5.90)$ | $(5.68,5.72,5.90)$
| $n=15$ |
RR${}_{2,3}-\gamma^{2}$ | $0.5$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$ | $(2.12,2.12,2.13)$
$0.7$ | $(4.12,4.08,4.19)$ | $(4.12,4.08,4.18)$ | $(4.12,4.08,4.18)$ | $(4.12,4.08,4.18)$ | $(4.12,4.08,4.18)$
$0.8$ | $(19.82,19.40,20.05)$ | $(19.83,19.41,19.99)$ | $(19.86,19.45,20.00)$ | $(19.85,19.45,20.00)$ | $(19.84,19.43,19.98)$
$1.3$ | $(9.68,9.83,10.47)$ | $(9.68,9.82,10.43)$ | $(9.68,9.82,10.42)$ | $(9.68,9.82,10.42)$ | $(9.68,9.82,10.41)$
$1.5$ | $(3.32,3.36,3.54)$ | $(3.32,3.36,3.53)$ | $(3.32,3.36,3.52)$ | $(3.32,3.36,3.52)$ | $(3.32,3.36,3.52)$
$2.0$ | $(2.13,2.14,2.18)$ | $(2.13,2.14,2.17)$ | $(2.13,2.14,2.17)$ | $(2.13,2.14,2.17)$ | $(2.13,2.14,2.17)$
RR${}_{3,4}-\gamma^{2}$ | $0.5$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$ | $(3.02,3.02,3.03)$
$0.7$ | $(4.02,4.02,4.10)$ | $(4.02,4.02,4.10)$ | $(4.01,4.02,4.09)$ | $(4.02,4.02,4.09)$ | $(4.01,4.02,4.09)$
$0.8$ | $(13.91,13.92,14.46)$ | $(13.90,13.92,14.44)$ | $(13.90,13.91,14.42)$ | $(13.92,13.90,14.42)$ | $(13.91,13.90,14.42)$
$1.3$ | $(9.78,9.91,10.47)$ | $(9.78,9.90,10.44)$ | $(9.78,9.90,10.43)$ | $(9.78,9.90,10.42)$ | $(9.78,9.90,10.42)$
$1.5$ | $(4.10,4.14,4.29)$ | $(4.10,4.13,4.28)$ | $(4.10,4.13,4.28)$ | $(4.10,4.13,4.28)$ | $(4.10,4.13,4.28)$
$2.0$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$ | $(3.09,3.10,3.13)$
RR${}_{4,5}-\gamma^{2}$ | $0.5$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$ | $(4.01,4.01,4.01)$
$0.7$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$ | $(4.58,4.59,4.64)$
$0.8$ | $(12.09,12.14,12.60)$ | $(12.09,12.14,12.57)$ | $(12.09,12.14,12.57)$ | $(12.09,12.14,12.56)$ | $(12.09,12.14,12.56)$
$1.3$ | $(10.37,10.49,11.02)$ | $(10.37,10.48,10.99)$ | $(10.37,10.48,10.98)$ | $(10.37,10.48,10.98)$ | $(10.37,10.48,10.98)$
$1.5$ | $(4.97,5.00,5.15)$ | $(4.97,5.00,5.14)$ | $(4.97,5.00,5.14)$ | $(4.97,5.00,5.13)$ | $(4.97,5.00,5.13)$
$2.0$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$ | $(4.07,4.08,4.10)$

Table 7: The $ARL$ values of the RR${}_{r,s}-\gamma^{2}$ control charts in the presence of measurement errors for $\gamma_{0}=0.05$ (left side), $\gamma_{0}=0.1$ (middle) and $\gamma_{0}=0.2$ (right side), and for different values of $m$, $\tau$, $n$, $\eta=0.28$, $\theta=0.05$, $B=1$. $i$ | $\bar{X}^{*}_{i}$ | $S^{*}_{i}$ | $\hat{\gamma}$ | $\hat{\gamma}^{*2}$
---|---|---|---|---
1 | 906.4 | 476.0 | 0.525 | 0.27563
2 | 805.1 | 493.9 | 0.614 | 0.37700
3 | 1187.2 | 1105.9 | 0.932 | 0.86862
4 | 663.4 | 304.8 | 0.459 | 0.21068
5 | 1012.1 | 367.4 | 0.363 | 0.13177
6 | 863.2 | 350.4 | 0.406 | 0.16484
7 | 1561.0 | 1562.2 | 1.058 | 1.11936
8 | 697.1 | 253.2 | 0.363 | 0.13177
9 | 1024.6 | 120.9 | 0.118 | 0.01392
10 | 355.3 | 235.2 | 0.662 | 0.43824
11 | 485.6 | 106.5 | 0.219 | 0.04796
12 | 1224.3 | 915.4 | 0.748 | 0.55950
13 | 1365.0 | 1051.6 | 0.770 | 0.59290
14 | 704.0 | 449.7 | 0.639 | 0.40832
15 | 1584.7 | 1050.8 | 0.663 | 0.43957
16 | 1130.0 | 680.6 | 0.602 | 0.36240
17 | 824.7 | 393.5 | 0.477 | 0.22753
18 | 921.2 | 391.6 | 0.425 | 0.18062
19 | 870.3 | 730.0 | 0.839 | 0.70392
20 | 1068.3 | 150.8 | 0.141 | 0.01988
Table 8: Illustrative example of Phase II dataset.