A New $p$-Control Chart with Measurement Error Correction

Li-Pang Chen111Email<EMAIL_ADDRESS>and Su-Fen Yang222Email:
<EMAIL_ADDRESS>(Corresponding Author).

Department of Statistics, National Chengchi University

Abstract

Control charts are important tools to monitor quality of products. One of
useful applications is to monitor the proportion of non-conforming products.
However, in practical applications, measurement error is ubiquitous and may
occur due to false records or misclassification, which makes the observed
proportion different from the underlying true proportion. It is also well-
known that ignoring measurement error effects provides biases, and is expected
that the resulting control charts may incur wrong detection. In this paper, we
study this important problem and propose a valid method to correct for
measurement error effects and obtain error-eliminated control chart for the
proportion of non-conforming products. In addition, unlike traditional
approaches, the corrected EWMA $p$-control chart provides asymmetric control
limits and is flexible to handle the data with small sample size. Numerical
results are conducted to justify the validity of the corrected EWMA
$p$-control chart and verify the necessity of measurement error correction.

Keywords: Asymmetric control limits; average run length; $p$-control charts;
error elimination; misclassification; monitoring.

Short title: SPC with misclassification

##  1 Introduction

Statistical process control (SPC) is an important tool in industrial
statistics and is useful in monitoring the quality of products. The main
interest is to develop control charts based on in-control (IC) statistic, and
then use them to monitor and detect out-of-control (OC) process parameters. To
do this, one may focus on location process or dispersion process. For example,
as summarized in Qiu (2013), many classical methods have been developed under
the parametric settings, including Shewhart charts, exponentially weighted
moving average (EWMA) charts, and cumulative sum (CUSUM) charts. Without
imposing parametric assumptions, distribution free settings have been popular
in recent years and relevant approaches have been developed in recent years,
such as EWMA mean charts (e.g. Yang 2016), variability monitoring (e.g., Yang
and Arnold 2016a; Yang and Wu 2017), and likelihood ratio-based EWMA control
charts (e.g., Zou and Tsung 2010). In addition, nonparametric methods have
been explored as well. For example, Chen et al. (2022) employed the kernel
estimation method to construct the control region based on location and
dispersion processes simultaneously.

In practice, noisy data usually exist and are inevitable in producing
products. A typical phenomenon in manufacturer factories is measurement error,
which reflects a fact that observed data are different from underlying true
ones that are unobservable. In the presence of continuous random variables,
some existing works have examined impacts of measurement error to different
kinds of control charts, such as EWMA charts (e.g., Maravelakis et al. 2004;
Asif et al. 2020; Nguyen et al. 2020; Tran et al. 2021), Shewhart control
charts (e.g., Nguyen et al. 2020; Linna and Woodall 2001), multivariate
process variability (e.g., Huwang and Hung 2007), first-order autoregressive
model (e.g., Shongwe et al. 2020), and multivariate control charts (e.g.,
Linna et al. 2001). However, while those works discussed measurement error
effects on control limits, to the best of knowledge, none of them provided
suitable strategies to correct for measurement error effects and adjust
control limits.

In addition to continuous random variables, sometimes we are interested in the
number/proportion of (non-)confirming products. Such a scenario refers binary
random variables. Under the assumption of parametric distributions, the
strategies of Shewhart charts have been adopted to construct $p$-control
charts (e.g., Qiu 2014, Section 3.3). Some relevant extensions have also been
explored, such as quasi ARL‐unbiased $p$‐charts based on a heuristic method
(e.g., Argoti and Carrión‐García 2019), the risk adjustment control charts for
categorical random variables (e.g., Sparks 2017), and the general framework of
distribution free settings (e.g., Yang and Arnold 2016b; Aslam et al. 2019).
Some applications of $p$-control charts have also been discussed. For example,
Shu and Wu (2010) applied $p$-control charts to monitor imprecise fraction of
conforming items. Bourke (2006) extended the $np$ chart for detecting upward
shifts in fraction defective. While the development of $p$-control chart has
been widely discussed, however, there are several critical concerns in
applications. First, similar to measurement error in continuous random
variables, it is possible to encounter measurement error in binary random
variables, which refers misclassification (e.g., Yi 2017, Section 2.6; Chen
and Yi 2021). More specifically, when producing products in factories,
sometimes they may be falsely detected as confirmed (or non-confirmed) due to
imprecise measurement equipment or human-made mistakes. Because of such a
misclassification, the observed status is different from what it should be.
While measurement error in continuous random variables has been widely
discussed, misclassification problem in SPC has been rarely discussed, and
strategies for correction of measurement error effects are unavailable. The
second concern is the small sample size in each monitoring time. It is known
that, by the central limit theorem, the estimated proportion follows normal
distributions when the sample size goes to the infinity. However, in the
presence of small sample size, the sampling distribution is unknown, and thus,
the construction of control charts is unknown.

Due to those concerns, in this paper we focus on SPC for binary random
variables subject to misclassification. We particularly focus on phase II
scenario where the parameter is assumed to be known. Under small sample sizes,
we employ EWMA charts to construct the asymmetric control charts. To deal with
measurement error effects, we propose the “corrected” proportion of non-
confirming products, so that the corresponding control limits can be adjusted
accordingly. To assess the performance of the corrected EWMA $p$-control
chart, we compare the corrected EWMA $p$-control chart with the naive
(uncorrected) EWMA $p$-control chart that simply adopt error-contaminated
random variables. Numerical experiments show that the corrected EWMA
$p$-control chart can precisely detect OC proportion.

The remainder is organized as follows. In Section 2, we introduce the data
structure and the EWMA $p$-control chart to monitor non-confirming products.
In addition, measurement error to binary random variables is introduced. In
Section 3, we propose a valid estimation procedure to correct the measurement
error effects and adopt the EWMA statistic to construct the asymmetric
corrected EWMA $p$-control chart. Moreover, we present in-control average run
length (ARL0) and out-of-control average run length (ARL1) to illustrate the
out-of-control detection performance. Empirical studies, including simulation
results and real data analysis, are provided in Sections 4 and 5,
respectively. We conclude the article with discussions in Section 6. The R
programming code for the implementation is available in the Github:
https://github.com/lchen723/SPC-ME-R-code.git.

##  2 Data Structure and Mismeasurement

In this section, we introduce the construction of the control chart with
relevant notation, and discuss the issue of measurement error in binary random
variables.

###  2.1 Binary Variables and Construction of The p-Control Chart

Let $T$ denote the monitoring time. For each time $t=1,\cdots,T$, there are
$n$ subjects. Let $X_{it}$ for $i=1,\cdots,n$ and $t=1,\cdots,T$ denote the
independent and identically distributed (i.i.d.) binary random variable with
outcome $0$ or $1$, where $X_{it}=1$ represents a non-conforming product,
while $X_{it}=0$ stands for a conforming product. The main interest is to
monitor the proportion of non-conforming products for in-control process.

Let $p_{0}\triangleq P(X_{it}=1)$ be the parameter of proportion of non-
conforming products for in-control process, and let $q_{0}\triangleq
1-p_{0}=P(X_{it}=0)$ denote the probability of conforming products for in-
control process. Under in-control $n$ subjects for $T$ monitoring times, we
define the sample proportion
$\widehat{p}_{0,t}\triangleq\frac{1}{n}\sum\limits_{i=1}^{n}X_{it}$.

To build up the reliable control chart, one may employ the exponentially
weighted moving average (EWMA) control chart, which is constructed based on a
weighted average of in-control sample proportion at the current time point. In
addition, as commented in Yang (2016), the EWMA chart is useful in monitoring
since it is sensitive in detecting small shifts in process parameters.

Based on the binary random variable and the sample proportion
$\widehat{p}_{0,t}$, the in-control EWMA statistic is given by

$\displaystyle\text{EWMA}_{0,t}=\lambda\widehat{p}_{0,t}+(1-\lambda)\text{EWMA}_{0,t-1},$
(1)

where $\text{EWMA}_{0,0}=p_{0}$ with $p_{0}$ being the in-control proportion,
and $\lambda\in(0,1]$ is a smoothing parameter. Moreover, when the monitoring
time goes by, the OC non-conforming proportion is non-decreasing and is
greater than IC non-conforming proportion. Thus, in applications, it is
reasonable to set the lower control limit (LCL) to be zero, and the upper
control limit (UCL) is specified as

$\displaystyle\text{UCL}=p_{0}+L\sqrt{\frac{p_{0}(1-p_{0})\lambda\\{1-(1-\lambda)^{2t}\\}}{n(2-\lambda)}}.$

Now, and hereafter, we call this strategy the EWMA $p$-control chart.

###  2.2 Misclassification

In practice, variables are often subject to mismeasurement. That is, instead
of collecting unobserved variables $X_{it}$, sometimes we have only the
observed or surrogate version of $X_{it}$, denoted by $X_{it}^{\ast}$. The
relationship between unobserved and observed variables
$X_{it},X_{it}^{\ast}\in\\{0,1\\}$ can be characterized as
$\pi_{kl}=P\big{(}X_{it}^{\ast}=k|X_{it}=l\big{)}$ for $k,l=0,1$.
Specifically, $\pi_{00}$ and $\pi_{11}$, which are called classification
probability, indicate that both $X_{it}$ and $X_{it}^{\ast}$ are either
conforming or non-conforming products; on the other hand, $\pi_{01}$ or
$\pi_{10}$ shows that the unobserved variable is opposite to the observed
variable which is caused by measurement error. In this case, we call
$\pi_{10}$ and $\pi_{01}$ as misclassification probability (e.g., Carroll et
al. 2006; Chen and Yi 2021). To further understand the concept of
misclassification, we can think of $X_{it}$ as the true status of product with
$X_{it}=1$ being non-conforming and 0 otherwise, which is unknown. On the
other hand, $X_{it}^{\ast}$ is understood as the observed status of product
that is recorded by factory’s staffs. Therefore, $\pi_{01}$ (or $\pi_{10}$) is
interpreted as the probability that a non-conforming (or conforming) product
is falsely recorded as conforming (or non-conforming).

Let $p_{0}^{\ast}=P(X_{it}^{\ast}=1)$ and
$q_{0}^{\ast}=1-p_{0}^{\ast}=P(X_{it}^{\ast}=0)$ denote the in-control
surrogate version of $p_{0}$ and $q_{0}$, respectively. To build up the
relationship of $p_{0}^{\ast}$ and $p_{0}$, we apply the technique of law of
total probability and obtain

$\displaystyle p_{0}^{\ast}=\pi_{11}p_{0}+\pi_{10}q_{0}.$ (2)

Similarly, $q_{0}^{\ast}$ and $q_{0}$ can be characterized as

$\displaystyle q_{0}^{\ast}=\pi_{01}p_{0}+\pi_{00}q_{0}.$ (3)

The matrix form of (2) and (3) is given by

$\displaystyle\left(\begin{array}[]{c}p_{0}^{\ast}\\\
q_{0}^{\ast}\end{array}\right)=\boldsymbol{\Pi}\left(\begin{array}[]{c}p_{0}\\\
q_{0}\end{array}\right),$ (8)

where $\boldsymbol{\Pi}=\left(\begin{array}[]{cc}\pi_{11}&\pi_{10}\\\
\pi_{01}&\pi_{00}\end{array}\right)$ is the $2\times 2$ (mis)classification
matrix and the sum of elements in each row is equal to one, i.e.,
$\pi_{11}+\pi_{10}=1$ and $\pi_{01}+\pi_{00}=1$. Following the similar
discussion in Chen and Yi (2021), we assume that $\boldsymbol{\Pi}$ has the
spectral decomposition
$\boldsymbol{\Pi}=\boldsymbol{\Omega}\mathbf{D}\boldsymbol{\Omega}^{-1}$,
where $\mathbf{D}$ is the diagonal matrix with diagonal elements being the
eigenvalues of $\boldsymbol{\Pi}$, and $\boldsymbol{\Omega}$ is the
corresponding matrix of eigenvectors.

Since our main target is to monitor $p_{0}$, however, from the equality (8),
we observe that the surrogate version $p_{0}^{\ast}$ is no longer equal to
$p_{0}$ if misclassification occurs, i.e., $\pi_{10}\neq 0$ or $\pi_{01}\neq
0$. In particular, when $\pi_{10}$ or $\pi_{01}$ become larger, $p_{0}^{\ast}$
and $q_{0}^{\ast}$ are different from $p_{0}$ and $q_{0}$. Therefore, with the
availability of $X_{it}^{\ast}$, the sample proportion
$\widehat{p}_{0,t}^{\ast}=\frac{1}{n}\sum\limits_{i=1}^{n}X_{it}^{\ast}$ and
$\widehat{q}_{0,t}^{\ast}=1-\widehat{p}_{0,t}^{\ast}$ have biases for $p_{0}$
and $q_{0}$, and the corresponding control limits, determined by (1) with
$\widehat{p}_{0,t}$ replaced by $\widehat{p}_{0,t}^{\ast}$, may incur wrong
detection.

###  2.3 Determination of Classification Matrix

While the probability of observed non-conforming or conforming products can be
expressed as (8), the (mis)classification matrix $\boldsymbol{\Pi}$ is usually
unknown because of involvement of the unobserved variable $X_{it}$. Therefore,
to develop the method, we consider sensitivity analyses (e.g., Chen 2020; Chen
and Yi 2021), whose purpose is to specify different values of
$\boldsymbol{\Pi}$ to understand how mismeasurement effects may affect
inference results and control charts, and it is usually employed when
additional information is unavailable.

Since the specification of $\boldsymbol{\Pi}$ may be not unique, in this paper
we consider the relative ratio (RR) to specify misclassification
probabilities. Specifically, let the relative ratio under $X_{it}=1$ be

$\displaystyle\text{RR}_{1}=\frac{\pi_{11}}{1-\pi_{11}},$

where $\text{RR}_{1}$ is in an interval $[0,\infty)$. A larger $\text{RR}_{1}$
indicates a higher probability of correct classification. Then given
$\text{RR}_{1}$, $\pi_{11}$ and $\pi_{01}$ can be derived as
$\pi_{11}=\frac{\text{RR}_{1}}{1+\text{RR}_{1}}$ and
$\pi_{01}=\frac{1}{1+\text{RR}_{1}}$, respectively. Similarly, under
$X_{it}=0$, the relative ratio is given by

$\displaystyle\text{RR}_{0}=\frac{\pi_{00}}{1-\pi_{00}}.$

Then for a specified $\text{RR}_{0}$, $\pi_{00}$ and $\pi_{10}$ are given by
$\pi_{00}=\frac{\text{RR}_{0}}{1+\text{RR}_{0}}$ and
$\pi_{10}=\frac{1}{1+\text{RR}_{0}}$, respectively. Therefore,
$\boldsymbol{\Pi}$ can be obtained.

##  3 Methodology

###  3.1 Corrected Control Limits under Expectation

Inspired by (8), an intuitive approach is to take the inverse matrix
$\boldsymbol{\Pi}^{-1}$ to both sides of (8), yielding

$\displaystyle\boldsymbol{\Pi}^{-1}\left(\begin{array}[]{c}p_{0}^{\ast}\\\
q_{0}^{\ast}\end{array}\right)=\left(\begin{array}[]{c}p_{0}\\\
q_{0}\end{array}\right).$ (13)

It suggests that $\boldsymbol{\Pi}^{-1}$ is regarded as the term to correct
for error effects in $p_{0}^{\ast}$ and $q_{0}^{\ast}$, and the left-hand-side
of (13) can reflect $p_{0}$ and $q_{0}$.

Specifically, from (13), $p_{0}$ can be expressed by

$\displaystyle p_{0}$ $\displaystyle=$
$\displaystyle\frac{\pi_{00}p_{0}^{\ast}-\pi_{10}q_{0}^{\ast}}{\pi_{11}\pi_{00}-\pi_{10}\pi_{01}}$
(14) $\displaystyle=$
$\displaystyle\frac{(1-\pi_{10})p_{0}^{\ast}-\pi_{10}q_{0}^{\ast}}{(1-\pi_{10})(1-\pi_{01})-\pi_{10}\pi_{01}}$
$\displaystyle=$
$\displaystyle\frac{p_{0}^{\ast}-\pi_{10}}{1-\pi_{10}-\pi_{01}},$

where the first equality is due to the manipulation of inverse matrix, the
second equality is due to the property of $\boldsymbol{\Pi}$, and the last
step comes from the fact $p_{0}^{\ast}+q_{0}^{\ast}=1$. Consequently, (14)
suggests that the “corrected” proportion of non-conforming products, denoted
as $p_{0}^{\ast\ast}$, is defined as the right-hand-side of (14). That is,

$\displaystyle
p_{0}^{\ast\ast}\triangleq\frac{p_{0}^{\ast}-\pi_{10}}{1-\pi_{10}-\pi_{01}}.$
(15)

Thus, it suggests the corrected random variable be

$\displaystyle
X_{it}^{\ast\ast}\triangleq\frac{X_{it}^{\ast}-\pi_{10}}{1-\pi_{10}-\pi_{01}}.$

Thus, the corrected sample proportion is denoted as
$\widehat{p}_{0,t}^{\ast\ast}=\frac{1}{n}\sum\limits_{i=1}^{n}X_{it}^{\ast\ast}$.

###  3.2 An Error Corrected Asymmetric EWMA $p$-Control Chart

As discussed in Section 2.1, we can take the EWMA as the monitoring statistic
and then develop the control chart. To address this, we apply (1) with
$\widehat{p}_{0,t}$ replaced by $\widehat{p}_{0,t}^{\ast\ast}$ to obtain the
corrected EWMA statistics, which is given by

$\displaystyle\text{EWMA}_{0,t}^{\ast\ast}=\lambda\widehat{p}_{0,t}^{\ast\ast}+(1-\lambda)\text{EWMA}_{0,t-1}^{\ast\ast}$
(16)

for $t=1,\cdots,T$, where $\lambda\in(0,1]$ is a smoothing parameter.

To construct the control chart, we require the expectation and variance of
(16) as stated in the following theorem.

###### Theorem 3.1

Under the monitoring statistic (16), we have

  * (a)

$E\left\\{\text{EWMA}_{0,t}^{\ast\ast}\right\\}=p_{0}$;

  * (b)

$\text{var}\left\\{\text{EWMA}_{0,t}^{\ast\ast}\right\\}=\frac{p_{0}^{\ast}(1-p_{0}^{\ast})\lambda\\{1-(1-\lambda)^{2t}\\}}{n(1-\pi_{10}-\pi_{01})^{2}(2-\lambda)}$.

Under Theorem 3.1, the corrected UCL and LCL are given by

$\displaystyle\text{UCL}^{\ast\ast}=p_{0}^{\ast\ast}+L^{\ast\ast}\sqrt{\frac{p_{0}^{\ast}(1-p_{0}^{\ast})\lambda\\{1-(1-\lambda)^{2t}\\}}{n(1-\pi_{10}-\pi_{01})^{2}(2-\lambda)}},\
\ \text{and}\ \ \text{LCL}^{\ast\ast}=0.$ (17)

We call (17) the corrected EWMA $p$-control chart.

###### Remark 3.1

  1. 1.

If the interest is $np$ charts, then simply multiplying $n$ to (17) yields the
desired result.

  2. 2.

If sample sizes in different subgroup are different from each other, then the
sample size $n$ in (17) can be replaced by $n_{k}$ for the $k$th subgroup.

###### Remark 3.2

(Naive EWMA statistic and $p$-control chart)

Under the observed random variable without error correction, we can also
construct the EWMA chart. Specifically, let

$\displaystyle\text{EWMA}_{0,t}^{\ast}=\lambda\widehat{p}_{0,t}^{\ast}+(1-\lambda)\text{EWMA}_{0,t-1}^{\ast}$
(18)

denote the naive EWMA statistic, and we define

$\displaystyle\text{UCL}^{\ast}=p_{0}^{\ast}+L^{\ast}\sqrt{\frac{p_{0}^{\ast}(1-p_{0}^{\ast})\lambda\\{1-(1-\lambda)^{2t}\\}}{n(2-\lambda)}},\
\ \text{LCL}^{\ast}=0$ (19)

as the naive EWMA $p$-control chart.

###  3.3 Examination of ARL

In the framework of statistical process control, to assess the performance of
$p$-control charts, we usually examine the average run length (ARL).

Let $\text{ARL}_{0}$ denote the in-control (IC) ARL, which reflects the mean
of the run length when the process is IC. In addition, let $\text{ARL}_{1}$ be
the out-of-control (OC) ARL, which gives the average number of samples
collected from the time of shift occurrence to the time of signal.

The goal is to use (17) to compute ARL0 and ARL1. Unlike the Shewhart chart
where ARL0 and ARL1 can reflect Type I and II errors, respectively, the
challenge is that the EWMA statistic has no such a property since it is
dependent. In addition, the coefficient of control limit is usually unknown
and the control limit is asymmetric. Therefore, to deal with those issues, we
first fix ARL0 at a given level, and then the calculate the coefficient of
control limit $L^{\ast\ast}$. After that, based on the constructed EWMA chart,
we compute ARL1. Detailed computational algorithms are placed in the following
two subsections, and similar strategy can be applied to the naive EWMA
$p$-control chart (19).

####  3.3.1 Determination of Critical Values and In-Control ARL

We first fix $\text{ARL}_{0}$ whose typical choices include 200 or 370. In
addition, under the error-prone data, we can use (8) to calculate
$p_{0}^{\ast}$. To address measurement error effects, we employ (15) to derive
the corrected IC proportion $p_{0}^{\ast\ast}$.

Given a range $(a,b)$ with user-specific values $a=0.01$ and $b=10$ for
$L^{\ast\ast}$, we aim to find the optimal value of $L^{\ast\ast}$ that
satisfies a given ARL0. The strategy to determine $L^{\ast\ast}$ is the Monte
Carlo method with repetition $M$. Under the $m$th Monte Carlo step, we
evaluate (16) and $\text{UCL}^{\ast\ast}$ in (17) for every $t=1,2,\cdots,T$
based on a proportion determined by IC samples. Let $t_{0,m}$ denote a run
length, which is defined as a value $t$ such that
$\text{EWMA}_{0,t}^{\ast\ast}\geq\text{UCL}^{\ast\ast}$. Finally, under $M$
repetitions in the Monte Carlo procedure, the estimated average run length
under IC samples is given by
$\widehat{\text{ARL}}_{0}\triangleq\frac{1}{M}\sum\limits_{m=1}^{M}t_{0,m}$,
and thus, $L^{\ast\ast}\in(a,b)$ can be determined by minimizing
$\Big{|}\text{ARL}_{0}-\widehat{\text{ARL}}_{0}\Big{|}<1$. The pseudo-code of
computational procedure for critical value $L^{\ast\ast}$ and ARL0 is
presented in Algorithm 1.

####  3.3.2 Determination of Out-of Control ARL

Let $p_{1}$ denote the OC process proportion that is determined by re-scaled
proportion $p_{0}$. Moreover, in the presence of measurement error, we have
the misclassified OC proportion $p_{1}^{\ast}$, which can be characterized
with respect to $p_{1}$ by the similar relationship of (8). To correct for
measurement error effects and recover misclassified proportion to the true
one, we again employ (15) to define the adjusted OC proportion, and denote it
as $p_{1}^{\ast\ast}$.

After adjusting the OC proportion, the next goal is to examine the constructed
control chart and compute ARL1. Following the similar strategy in Section
3.3.1, we employ the Monte Carlo method to derive ARL1 based on the
constructed EWMA control chart. For the $m$th step in the Monte Carlo
procedure, we use (16) to compute EWMA based on OC proportion for each monitor
time $t$. After that, let $t_{1,m}$ denote an OC run length, which is given by
a value $t$ that satisfies
$\text{EWMA}_{1,t}^{\ast\ast}\geq\text{UCL}^{\ast\ast}$ with
$\text{UCL}^{\ast\ast}$ being determined in Algorithm 1. Consequently, under
$M$ repetitions in the Monte Carlo procedure, the estimated average run length
under OC samples is given by
$\widehat{\text{ARL}}_{1}=\frac{1}{M}\sum\limits_{m=1}^{M}t_{1,m}$. The
pseudo-code of computation of ARL1 is presented in Algorithm 2.

##  4 Simulation Studies

###  4.1 Simulation Setup

Let $n$ denote the sample size in each monitoring time, where $n=5,10,15,$ and
$20$. Let $T$ be the monitoring time that is set as $T=5000$. For
$i=1,\cdots,n$ and $t=1,\cdots,T$, let the true IC data $X_{it}$ be generated
from the Bernoulli distribution with the IC probability $p_{0}$ that is
specified as 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5. Moreover,
for the OC samples, the proportion $p_{1}$ is defined as
$p_{1}\triangleq(1+\delta)p_{0}$ with $\delta$ being specified as 0.1 and 0.2.

For the misclassification model (8), we specify $\pi_{00}=\pi_{11}=\pi$ and
$\pi_{10}=\pi_{01}=1-\pi$ with $\pi=0.95$ and $0.99$, yielding reliability
ratios $\text{RR}_{1}=\text{RR}_{0}=0.474,0.487,0.498$, respectively. Based on
the misclassification model (8), we generate the error-prone binary random
variable $X^{\ast}$. We denote $p_{0}^{\ast}$ and $p_{1}^{\ast}$ as the
corresponding error-prone IC and OC probabilities, respectively.

To implement the corrected EWMA $p$-control chart, we adopt (15) to calculate
the corrected probability $p_{0}^{\ast\ast}$, and then employ (16) to
construct the EWMA statistics with smoothing parameters being $\lambda=0.05$
or $0.2$. Finally, 10000 simulations are run for all settings.

###  4.2 Simulation Results

To assess the OC detection performance of the corrected EWMA $p$-control
chart, we also investigate the naive method in Remark 3.2 and (1) by assuming
the existence of the true $X$. While the common choice of ARL0 can be 370 or
200, here we only explore 370 because the result under ARL${}_{0}=200$ has the
similar pattern to that based on ARL${}_{0}=370$.

We first present numerical results for coefficients of the control limit and
UCLs based on given $\pi$ and $\lambda$ in Tables 1-4. We observe that UCL
decreases when $n$ becomes large regardless of the methods. In the presence of
misclassification, we observe that UCL∗ increases when misclassification is
severe (i.e., $\pi$ becomes small); on the contrary, it is interesting to see
that values of UCL∗∗ determined by the corrected EWMA $p$-control chart keep
remained and are the same as UCL based on the true $X$. In addition, values of
UCL∗ are greater than others, suggesting wider control limits determined by
error-prone variables. On the other hand, with $\pi$ fixed, values of UCL will
increase when $\lambda$ is increasing regardless of the implementation of
methods. Regarding the results of coefficients of the control limit $L$, we
observe that $L^{\ast\ast}$ is smaller than $L^{\ast}$ for most settings, and
values of $L^{\ast\ast}$ have the same decreasing pattern as values of $\pi$
and $\lambda$ do.

After obtaining estimated control limits, we next assess the estimation
results of ARL1. Numerical results under ARL${}_{0}=370$ are placed in Tables
LABEL:ARL1_370_0.05_0.95-LABEL:ARL1_370_0.2_0.99, where ARL1 with superscripts
$\ast$ and $\ast\ast$ represent the naive and proposed methods, respectively,
and the results for the true $X$ are recorded as ARL1 without the superscript.
We observe that all values decease when $n$, $p_{1}$, or $\delta$ increase. In
general, we can see that values of ARL${}_{1}^{\ast}$ are always greater than
ARL1 and ARL${}_{1}^{\ast\ast}$, and differences become large as $\delta$ and
$n$ are small. It shows that the naive method is unsatisfactory to detect OC
proportion, because smaller values of ARL1 have better performance to detect
OC (e.g., Maravelakis et al. 2004). On the other hand, values of
ARL${}_{1}^{\ast\ast}$ are close to ARL1 under the true $X$. These results
suggest that (1) the necessity of measurement error correction, (2) the
correction of measurement error effects recover the control limit to that
determined by the true $X$, and makes the detection of OC be the same as that
of true $X$, (3) the corrected EWMA $p$-control chart is successful to handle
the scenario of small sample size and is robust in detecting OC proportion.

##  5 Application to Real Data Analysis

In this section, we apply the corrected EWMA $p$-control chart to the orange
juice data, which can be found in the R package qcr.

The primary interest of this study is the production of 6-oz cardboard cans
that frozen orange juice concentrate is packed in. These cans are formed on a
machine by spinning them from cardboard stock and attaching a metal bottom
panel. A can is then inspected to determine whether, when filled, the liquid
could possible leak either on the side seam or around the bottom joint. If
this situation occurs, then a can is considered non-conforming. The data were
collected as 30 samples of 50 cans each at half-hour intervals over a three-
shift period in which the machine was in continuous operation. From sample 15
used, a new batch of cardboard stock was punt into production. Sample 23 was
obtained when an inexperienced operator was temporarily assigned to the
machine. After the first 30 samples, a machine adjustment was made. Then
further 24 samples were taken from the process. Therefore, according to the
data description, we take the IC monitoring time as $T=24$ and assign the OC
monitoring time as 30. For each IC monitoring time, the sample size is $n=50$.

In applications, measurement error is ubiquitous in process monitoring. In
addition, Pendrill (2014) and Puydarrieux et al. (2019) pointed out that risks
arising from measurement are main concern in conformity assessment. In other
words, from the orange juice data, it is possible that factory’s staffs
falsely record non-conforming (or conforming) cans to be conforming (or non-
conforming). As a result, it is reasonable to assume that the collected data
are subject to measurement error, and the corresponding proportion is given by
$p_{0}^{\ast}=0.111$.

Since measurement error correction is crucial, we now adopt (8) to to
characterize measurement error and employ the corrected EWMA $p$-control chart
(15) to correct for measurement error effects. Since $\boldsymbol{\Pi}$ in (8)
is unknown and additional information is unavailable, we employ sensitivity
analyses to address different levels of measurement error effects.
Specifically, we follow Section 4.1 to specify $\pi_{00}=\pi_{11}=\pi$ and
$\pi_{10}=\pi_{01}=1-\pi$ with $\pi=0.95$ and $0.99$, which yield
$p_{0}^{\ast\ast}=0.068$ and $0.103$, respectively.

Now, for each fixed $\pi=0.95$ or $0.99$ and $\lambda=0.05$ or $0.2$, we adopt
(16) and (18) to compute the EWMA statistics, and apply Algorithm 1 to compute
$L$ and UCL based on a given ARL0, yielding the corrected or naive EWMA
$p$-control charts, respectively. For the choice of ARL0, we simply use 370
for exploration, and ARL${}_{0}=200$ gives the similar result.

We first report the estimation results for $L$ and UCL based on the naive and
corrected EWMA $p$-control charts in Table 9. We observe that coefficients $L$
and UCL values based on the naive EWMA $p$-control chart are larger than those
based on the corrected EWMA $p$-control chart regardless of choices of
$\lambda$, which indicates that the naive EWMA $p$-control chart gives wider
control limits than those given by the corrected EWMA $p$-control chart. In
addition, from the corrected EWMA $p$-control chart, we can see that both $L$
and UCL are decreasing when $\pi$ becomes small, which reflects a phenomenon
that the corrected control limits may be implicitly affected by the proportion
of misclassification. Moreover, we also display naive and corrected EWMA
$p$-control charts based on $\lambda=0.05$ and $0.2$ in Figures 1 and 2,
respectively, for visualization. We can see that all monitoring points are
under the upper control limit, except for the third point that is known as the
false alarm based on $\pi=0.95$ in Figure 2 it may show that severe
misclassification makes detection more sensitive.

Finally, based on the developed naive or corrected EWMA $p$-control charts, we
further detect OC process parameters, and the associated control charts under
$\lambda=0.05$ or $0.2$ and $\pi=0.95$ or $0.99$ are displayed in Figures 3
and 4, respectively. We observe that detected points under uncorrected or
corrected EWMA $p$-control charts have similar pattern, and values of EWMA
statistic become small as $\pi$ is decreasing. In addition, the naive EWMA
$p$-control chart seems to be sensitive to detect OC with the choice of
$\lambda$. In particular, when $\lambda=0.05$, the second point of the naive
EWMA $p$-control chart in Figure 3 is not detected as OC, while the corrected
EWMA $p$-control chart can successfully detect OC regardless of the choices of
$\lambda$ or $\pi$.

##  6 Discussion

Statistical process control has been an important tool to monitor products.
Some methods have been developed to construct control charts for binary and/or
continuous random variables. In applications, however, measurement error
exists due to imprecise operation systems or human-made mistakes, and ignoring
measurement error effects may cause wrong detection. While many research works
have been available to discuss measurement error effects on continuous random
variables, however, to the best of our knowledge, little attention focused on
binary random variables. Moreover, none of them provided suitable strategies
to adjust error effects. In this paper, we consider binary random variables
that can be used to describe non-conforming product, and we allow binary
random variables to be contaminated by misclassification. We proposed the
corrected proportion to adjust for measurement error effects, and then employ
the corrected EWMA $p$-control chart to obtain reliable and asymmetric control
chart under small sample size. Numerical results based on different setting
verify the validity of the corrected EWMA $p$-control chart.

One of the concerns in misclassification model (8) is the determination of
$\boldsymbol{\Pi}$. In our development, we employ sensitivity analyses to
explore the impact of measurement error effects by examining different levels
of misclassification probabilities. In the framework of measurement error
analysis, $\boldsymbol{\Pi}$ can be estimated if auxiliary information is
available. One of typical information is external validation data.
Specifically, suppose that $\mathcal{M}$ with $|\mathcal{M}|=n$ is the subject
set for the main study containing measurements
$\left\\{X_{it}^{\ast}:i\in\mathcal{M},t=1,\cdots,T\right\\}$ and let
$\mathcal{V}$ with $|\mathcal{V}|=m$ denote the subject set for the external
validation study containing measurements
$\left\\{\left(X_{it},X_{it}^{\ast}\right):i\in\mathcal{V},t=1,\cdots,T\right\\}$,
where $\mathcal{M}$ and $\mathcal{V}$ do not overlap. Assume that the main
study and the validation study share the same model (8). With the availability
of external validation data $\mathcal{V}$, we have a $2\times 2$ confusion
table. Then the probability $\pi_{kl}$ for $k,l\in\\{0,1\\}$ can be estimated
by

$\displaystyle\widehat{\pi}_{kl}=\frac{\text{number of
}\left\\{X_{it}^{\ast}=k\ \text{and}\ X_{it}=l\right\\}\text{ for all
}i\in\mathcal{V}\text{ and }t=1,\cdots,T}{\text{number of
}\left\\{X_{it}=l\right\\}\text{ for all }i\in\mathcal{V}\text{ and
}t=1,\cdots,T}.$

Therefore, $\boldsymbol{\Pi}$ can be estimated by
$\widehat{\boldsymbol{\Pi}}=\big{[}\widehat{\pi}_{kl}\big{]}_{k,l\in\\{0,1\\}}$.

There are some possible extensions for this project. First of all, in addition
to the current strategy in Section 3.1 to address measurement error effects,
in the framework of measurement error analysis, simulation and extrapolation
(SIMEX) method (e.g., Chen 2020) is also a valid tool to measurement error
effects. It is a worth exploration in depth because no relevant work has been
available for SPC. Moreover, the corrected EWMA $p$-control chart can be
naturally extended to other settings. For example, as developed by Yang and
Arnold (2016a, 2016b) and Yang (2017), to address distribution free continuous
random variables and build up control chart, a common approach to translate
the continuous random variables to binary ones. In the presence of measurement
error, measurement error effects may affect the translated random variables
and associated proportion. Therefore, the corrected EWMA $p$-control chart and
the corrected proportion can be employed to deal with this issue. The other
interesting is about the profile monitoring (e.g., Qiu 2014, Chapter 10),
which aims to take auxiliary information as the covariates, and then build up
a regression model whose responses are the main interest to be monitored. In
the case of monitoring confirming product, we may consider logistic regression
models, and the similar strategy in Section 3.1 can be adopted to address this
concern. Detailed and deep explorations can be our next research projects in
the future.

Step 1: Given in-control $p_{0}$ and $\boldsymbol{\Pi}$, $p_{0}^{\ast}$ is
calculated by formula (8);

Step 2: Set $\lambda$, $n$, and a value of ARL0;

Step 3: Set $a<L^{\ast\ast}<b$ with $a=0.01$ and $b=10$ (say);

Step 4: Monte Carlo procedure:

for  _step $(m+1)$ with $m=1,2,\cdots,M$ and set $M=10001$ (say)_ do

Step 4.1: Let $\text{EWMA}_{0,0}^{\ast\ast}=p_{0}^{\ast\ast}$ and $t=1$;

Step 4.2: Based on the observed and misclassified $X_{it}^{\ast}$, calculate
$\widehat{p}_{0,t}^{\ast\ast}\triangleq\frac{1}{n}\sum\limits_{i=1}^{n}\frac{X_{it}^{\ast}-\pi_{10}}{1-\pi_{10}-\pi_{01}}$
;

  * •

If $t=1$,
$\text{EWMA}_{0,1}^{\ast\ast}=\lambda\widehat{p}_{0,1}^{\ast\ast}+(1-\lambda)\text{EWMA}_{0,0}^{\ast\ast}$;

  * •

If $t\neq 1$,
$\text{EWMA}_{0,t}^{\ast\ast}=\lambda\widehat{p}_{0,t}^{\ast\ast}+(1-\lambda)\text{EWMA}_{0,t-1}^{\ast\ast}$.

Step 4.3: Set UCL∗∗ and LCL∗∗ as values in (17) ;

  * •

If $\text{EWMA}_{0,t}^{\ast\ast}\geq\text{UCL}^{\ast\ast}$, then take
$t_{0,m}\triangleq t$ as a run length and $m\leftarrow m+1$.

Go to Step 1;

  * •

If $0<\text{EWMA}_{0,t}^{\ast\ast}\leq\text{UCL}^{\ast\ast}$, then
$t\leftarrow t+1$. Go to Step 2.

end for

Step 5:

Calculate $\frac{1}{M}\sum\limits_{m=1}^{M}t_{0,m}$ and take it as the
estimate of the nominal ARL0, which is denoted as $\widehat{\text{ARL}}_{0}$.
Finally, determine $L^{\ast\ast}$ by minimizing
$\Big{|}\text{ARL}_{0}-\widehat{\text{ARL}}_{0}\Big{|}<1$ subject to
$a<L^{\ast\ast}<b$.

Algorithm 1 Monte Carlo simulation to find coefficients of the control limit
and ARL0

Step 1:

Given out-of-control $p_{1}^{\ast}$, $\lambda$, $n$, a nominal ARL0, and
UCL∗∗;

* Step 2:

Based on $X_{it}^{\ast}$ in OC, calculate
$\widehat{p}_{1,t}^{\ast\ast}\triangleq\frac{1}{n}\sum\limits_{i=1}^{n}\frac{X_{it}^{\ast}-\pi_{10}}{1-\pi_{10}-\pi_{01}}$;

Step 3: Monte Carlo procedure:

for  _step $(m+1)$ with $m=1,2,\cdots,M$ and set $M=10001$ (say)_ do

Step 3.1: Let $\text{EWMA}_{1,0}^{\ast\ast}=p_{0}^{\ast\ast}$ and $t=1$;

Step 3.2: calculate out-of-control statistics, denoted
$\text{EWMA}_{1,t}^{\ast\ast}$:

  * •

If $t=1$,
$\text{EWMA}_{1,1}^{\ast\ast}=\lambda\widehat{p}_{1,1}^{\ast\ast}+(1-\lambda)\text{EWMA}_{1,0}^{\ast\ast}$;

  * •

If $t\neq 1$,
$\text{EWMA}_{1,t}^{\ast\ast}=\lambda\widehat{p}_{1,t}^{\ast\ast}+(1-\lambda)\text{EWMA}_{1,t-1}^{\ast\ast}$.

Step 3.3: Plot $\text{EWMA}_{1,t}^{\ast\ast}$ in the chart.

  * •

If $\text{EWMA}_{1,t}^{\ast\ast}\geq\text{UCL}^{\ast\ast}$, then take
$t_{1,m}\triangleq t$ as a run length and $m\leftarrow m+1$.

Go to Step 3.1;

  * •

If $\text{EWMA}_{1,t}^{\ast\ast}\leq\text{UCL}^{\ast\ast}$, then $t\leftarrow
t+1$. Go to Step 3.2.

end for

Step 4: Calculate the estimated ARL1, which is given by
$\widehat{\text{ARL}}_{1}=\frac{1}{M}\sum\limits_{m=1}^{M}t_{1,m}$.

Algorithm 2 Calculation for ARL1

Table 1: Simulation results ARL${}_{0}=370$, $\lambda=0.05$, and $\pi=0.95$. | $p_{0}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
---|---|---|---|---|---|---|---|---|---|---|---
| $p_{0}^{\ast}$ | 0.095 | 0.140 | 0.185 | 0.230 | 0.275 | 0.320 | 0.365 | 0.410 | 0.455 | 0.500
$n$ | $p_{0}^{\ast\ast}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
5 | $L$ | 2.463 | 2.355 | 2.290 | 2.281 | 2.260 | 2.242 | 2.222 | 2.205 | 2.195 | 2.185
| UCL | 0.088 | 0.151 | 0.209 | 0.265 | 0.320 | 0.374 | 0.426 | 0.477 | 0.528 | 0.578
| $L^{\ast}$ | 2.363 | 2.311 | 2.305 | 2.263 | 2.261 | 2.236 | 2.215 | 2.205 | 2.194 | 2.184
|  UCL∗ | 0.145 | 0.197 | 0.249 | 0.298 | 0.347 | 0.395 | 0.441 | 0.488 | 0.533 | 0.578
| $L^{\ast\ast}$ | 1.645 | 1.833 | 1.902 | 1.955 | 1.975 | 1.976 | 1.980 | 1.980 | 1.982 | 1.966
|  UCL∗∗ | 0.088 | 0.151 | 0.209 | 0.265 | 0.320 | 0.373 | 0.426 | 0.477 | 0.529 | 0.578
10 | $L$ | 2.374 | 2.295 | 2.270 | 2.258 | 2.230 | 2.221 | 2.211 | 2.202 | 2.185 | 2.182
| UCL | 0.076 | 0.135 | 0.191 | 0.246 | 0.299 | 0.352 | 0.403 | 0.455 | 0.505 | 0.555
| $L^{\ast}$ | 2.316 | 2.285 | 2.261 | 2.241 | 2.236 | 2.215 | 2.204 | 2.192 | 2.188 | 2.180
|  UCL∗ | 0.129 | 0.180 | 0.229 | 0.278 | 0.326 | 0.372 | 0.419 | 0.465 | 0.510 | 0.555
| $L^{\ast\ast}$ | 1.584 | 1.794 | 1.874 | 1.936 | 1.945 | 1.960 | 1.968 | 1.973 | 1.971 | 1.965
|  UCL∗∗ | 0.076 | 0.135 | 0.191 | 0.246 | 0.299 | 0.351 | 0.403 | 0.455 | 0.505 | 0.555
15 | $L$ | 2.343 | 2.270 | 2.260 | 2.243 | 2.230 | 2.210 | 2.201 | 2.192 | 2.180 | 2.179
| UCL | 0.071 | 0.128 | 0.183 | 0.237 | 0.290 | 0.342 | 0.393 | 0.444 | 0.495 | 0.545
| $L^{\ast}$ | 2.286 | 2.256 | 2.242 | 2.228 | 2.222 | 2.213 | 2.200 | 2.194 | 2.185 | 2.180
|  UCL∗ | 0.123 | 0.172 | 0.221 | 0.269 | 0.316 | 0.363 | 0.409 | 0.455 | 0.500 | 0.545
| $L^{\ast\ast}$ | 1.562 | 1.772 | 1.871 | 1.921 | 1.942 | 1.954 | 1.962 | 1.965 | 1.971 | 1.963
|  UCL∗∗ | 0.071 | 0.128 | 0.183 | 0.237 | 0.290 | 0.342 | 0.393 | 0.444 | 0.495 | 0.545
20 | $L$ | 2.323 | 2.267 | 2.247 | 2.236 | 2.216 | 2.206 | 2.202 | 2.191 | 2.180 | 2.178
| UCL | 0.068 | 0.124 | 0.179 | 0.232 | 0.284 | 0.336 | 0.388 | 0.438 | 0.489 | 0.539
| $L^{\ast}$ | 2.284 | 2.250 | 2.235 | 2.220 | 2.215 | 2.205 | 2.197 | 2.186 | 2.185 | 2.178
|  UCL∗ | 0.119 | 0.168 | 0.216 | 0.263 | 0.310 | 0.357 | 0.403 | 0.448 | 0.494 | 0.539
| $L^{\ast\ast}$ | 1.552 | 1.764 | 1.857 | 1.912 | 1.932 | 1.950 | 1.962 | 1.962 | 1.965 | 1.962
|  UCL∗∗ | 0.068 | 0.124 | 0.179 | 0.232 | 0.284 | 0.336 | 0.388 | 0.438 | 0.489 | 0.539
Table 2: Simulation results ARL${}_{0}=370$, $\lambda=0.05$, and $\pi=0.99$. | $p_{0}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
---|---|---|---|---|---|---|---|---|---|---|---
| $p_{0}^{\ast}$ | 0.059 | 0.108 | 0.157 | 0.206 | 0.255 | 0.304 | 0.353 | 0.402 | 0.451 | 0.500
$n$ | $p_{0}^{\ast\ast}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
5 | $L$ | 2.463 | 2.355 | 2.290 | 2.281 | 2.260 | 2.242 | 2.222 | 2.205 | 2.195 | 2.185
| UCL | 0.088 | 0.151 | 0.209 | 0.265 | 0.320 | 0.374 | 0.426 | 0.477 | 0.528 | 0.578
| $L^{\ast}$ | 2.432 | 2.341 | 2.306 | 2.276 | 2.257 | 2.237 | 2.216 | 2.203 | 2.195 | 2.184
|  UCL∗ | 0.100 | 0.160 | 0.217 | 0.272 | 0.325 | 0.378 | 0.429 | 0.479 | 0.529 | 0.578
| $L^{\ast\ast}$ | 2.228 | 2.228 | 2.205 | 2.215 | 2.200 | 2.185 | 2.170 | 2.158 | 2.154 | 2.139
|  UCL∗∗ | 0.088 | 0.151 | 0.209 | 0.265 | 0.320 | 0.373 | 0.426 | 0.477 | 0.528 | 0.578
10 | $L$ | 2.374 | 2.295 | 2.270 | 2.258 | 2.230 | 2.221 | 2.211 | 2.202 | 2.185 | 2.182
| UCL | 0.076 | 0.135 | 0.191 | 0.246 | 0.299 | 0.352 | 0.403 | 0.455 | 0.505 | 0.555
| $L^{\ast}$ | 2.361 | 2.295 | 2.257 | 2.250 | 2.230 | 2.215 | 2.206 | 2.198 | 2.190 | 2.182
|  UCL∗ | 0.087 | 0.144 | 0.199 | 0.252 | 0.304 | 0.356 | 0.406 | 0.457 | 0.506 | 0.555
| $L^{\ast\ast}$ | 2.144 | 2.183 | 2.182 | 2.190 | 2.169 | 2.167 | 2.157 | 2.153 | 2.144 | 2.137
|  UCL∗∗ | 0.076 | 0.135 | 0.191 | 0.246 | 0.299 | 0.352 | 0.403 | 0.455 | 0.505 | 0.555
15 | $L$ | 2.343 | 2.270 | 2.260 | 2.243 | 2.230 | 2.210 | 2.201 | 2.192 | 2.180 | 2.179
| UCL | 0.071 | 0.128 | 0.183 | 0.237 | 0.290 | 0.342 | 0.393 | 0.444 | 0.495 | 0.545
| $L^{\ast}$ | 2.324 | 2.267 | 2.251 | 2.232 | 2.225 | 2.216 | 2.206 | 2.196 | 2.187 | 2.180
|  UCL∗ | 0.082 | 0.137 | 0.191 | 0.243 | 0.295 | 0.346 | 0.397 | 0.447 | 0.496 | 0.545
| $L^{\ast\ast}$ | 2.115 | 2.156 | 2.172 | 2.172 | 2.166 | 2.158 | 2.151 | 2.149 | 2.144 | 2.137
|  UCL∗∗ | 0.071 | 0.128 | 0.183 | 0.237 | 0.290 | 0.342 | 0.393 | 0.444 | 0.495 | 0.545
20 | $L$ | 2.323 | 2.267 | 2.247 | 2.236 | 2.216 | 2.206 | 2.202 | 2.191 | 2.180 | 2.178
| UCL | 0.068 | 0.124 | 0.179 | 0.232 | 0.284 | 0.336 | 0.388 | 0.438 | 0.489 | 0.539
| $L^{\ast}$ | 2.314 | 2.267 | 2.247 | 2.228 | 2.218 | 2.211 | 2.204 | 2.193 | 2.185 | 2.178
|  UCL∗ | 0.079 | 0.133 | 0.186 | 0.238 | 0.290 | 0.340 | 0.391 | 0.440 | 0.490 | 0.539
| $L^{\ast\ast}$ | 2.102 | 2.144 | 2.159 | 2.167 | 2.157 | 2.156 | 2.151 | 2.149 | 2.144 | 2.137
|  UCL∗∗ | 0.068 | 0.124 | 0.179 | 0.232 | 0.284 | 0.336 | 0.388 | 0.438 | 0.489 | 0.539
Table 3: Simulation results ARL${}_{0}=370$, $\lambda=0.2$, and $\pi=0.95$. | $p_{0}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
---|---|---|---|---|---|---|---|---|---|---|---
| $p_{0}^{\ast}$ | 0.095 | 0.140 | 0.185 | 0.230 | 0.275 | 0.320 | 0.365 | 0.410 | 0.455 | 0.500
$n$ | $p_{0}^{\ast\ast}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
5 | $L$ | 3.336 | 3.068 | 2.946 | 2.862 | 2.793 | 2.740 | 2.693 | 2.644 | 2.618 | 2.589
| UCL | 0.158 | 0.237 | 0.307 | 0.371 | 0.430 | 0.487 | 0.541 | 0.593 | 0.644 | 0.693
| $L^{\ast}$ | 3.078 | 2.976 | 2.883 | 2.807 | 2.769 | 2.722 | 2.686 | 2.629 | 2.603 | 2.588
|  UCL∗ | 0.230 | 0.294 | 0.352 | 0.406 | 0.459 | 0.509 | 0.558 | 0.603 | 0.648 | 0.693
| $L^{\ast\ast}$ | 2.238 | 2.388 | 2.442 | 2.442 | 2.439 | 2.427 | 2.403 | 2.364 | 2.345 | 2.315
|  UCL∗∗ | 0.159 | 0.237 | 0.307 | 0.370 | 0.430 | 0.488 | 0.542 | 0.593 | 0.643 | 0.692
10 | $L$ | 3.138 | 2.945 | 2.855 | 2.794 | 2.745 | 2.713 | 2.675 | 2.641 | 2.616 | 2.586
| UCL | 0.122 | 0.193 | 0.257 | 0.318 | 0.375 | 0.431 | 0.484 | 0.536 | 0.587 | 0.636
| $L^{\ast}$ | 2.960 | 2.878 | 2.793 | 2.767 | 2.718 | 2.696 | 2.660 | 2.642 | 2.609 | 2.587
|  UCL∗ | 0.186 | 0.245 | 0.299 | 0.353 | 0.403 | 0.453 | 0.500 | 0.547 | 0.592 | 0.636
| $L^{\ast\ast}$ | 2.097 | 2.288 | 2.367 | 2.390 | 2.396 | 2.399 | 2.382 | 2.370 | 2.348 | 2.331
|  UCL∗∗ | 0.122 | 0.193 | 0.258 | 0.318 | 0.375 | 0.431 | 0.484 | 0.537 | 0.587 | 0.637
15 | $L$ | 3.044 | 2.893 | 2.814 | 2.768 | 2.723 | 2.687 | 2.661 | 2.637 | 2.616 | 2.585
| UCL | 0.107 | 0.175 | 0.236 | 0.295 | 0.351 | 0.406 | 0.459 | 0.511 | 0.562 | 0.611
| $L^{\ast}$ | 2.897 | 2.832 | 2.775 | 2.735 | 2.705 | 2.686 | 2.656 | 2.635 | 2.617 | 2.587
|  UCL∗ | 0.168 | 0.225 | 0.278 | 0.329 | 0.379 | 0.428 | 0.475 | 0.522 | 0.567 | 0.611
| $L^{\ast\ast}$ | 2.037 | 2.253 | 2.329 | 2.364 | 2.379 | 2.377 | 2.372 | 2.364 | 2.353 | 2.331
|  UCL∗∗ | 0.107 | 0.175 | 0.236 | 0.295 | 0.352 | 0.406 | 0.459 | 0.511 | 0.562 | 0.611
20 | $L$ | 2.980 | 2.858 | 2.790 | 2.748 | 2.710 | 2.678 | 2.655 | 2.635 | 2.614 | 2.585
| UCL | 0.098 | 0.164 | 0.224 | 0.282 | 0.337 | 0.391 | 0.444 | 0.496 | 0.547 | 0.596
| $L^{\ast}$ | 2.859 | 2.796 | 2.756 | 2.726 | 2.691 | 2.674 | 2.654 | 2.632 | 2.612 | 2.586
|  UCL∗ | 0.157 | 0.212 | 0.265 | 0.316 | 0.365 | 0.413 | 0.460 | 0.506 | 0.552 | 0.596
| $L^{\ast\ast}$ | 1.995 | 2.224 | 2.309 | 2.349 | 2.362 | 2.372 | 2.368 | 2.364 | 2.348 | 2.332
|  UCL∗∗ | 0.098 | 0.164 | 0.224 | 0.282 | 0.337 | 0.392 | 0.444 | 0.496 | 0.547 | 0.597
Table 4: Simulation results ARL${}_{0}=370$, $\lambda=0.2$, and $\pi=0.99$. | $p_{0}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
---|---|---|---|---|---|---|---|---|---|---|---
| $p_{0}^{\ast}$ | 0.059 | 0.108 | 0.157 | 0.206 | 0.255 | 0.304 | 0.353 | 0.402 | 0.451 | 0.500
$n$ | $p_{0}^{\ast\ast}$ | 0.050 | 0.100 | 0.150 | 0.200 | 0.250 | 0.300 | 0.350 | 0.400 | 0.450 | 0.500
5 | $L$ | 3.336 | 3.068 | 2.946 | 2.862 | 2.793 | 2.740 | 2.693 | 2.644 | 2.618 | 2.589
| UCL | 0.158 | 0.237 | 0.307 | 0.371 | 0.430 | 0.487 | 0.541 | 0.593 | 0.644 | 0.693
| $L^{\ast}$ | 3.257 | 3.061 | 2.935 | 2.842 | 2.792 | 2.748 | 2.693 | 2.633 | 2.603 | 2.588
|  UCL∗ | 0.173 | 0.250 | 0.316 | 0.377 | 0.436 | 0.492 | 0.545 | 0.594 | 0.644 | 0.693
| $L^{\ast\ast}$ | 3.032 | 2.911 | 2.836 | 2.769 | 2.718 | 2.680 | 2.637 | 2.584 | 2.553 | 2.520
|  UCL∗∗ | 0.159 | 0.237 | 0.307 | 0.370 | 0.430 | 0.488 | 0.542 | 0.593 | 0.643 | 0.692
10 | $L$ | 3.138 | 2.945 | 2.855 | 2.794 | 2.745 | 2.713 | 2.675 | 2.641 | 2.616 | 2.586
| UCL | 0.122 | 0.193 | 0.257 | 0.318 | 0.375 | 0.431 | 0.484 | 0.536 | 0.587 | 0.636
| $L^{\ast}$ | 3.095 | 2.913 | 2.845 | 2.789 | 2.741 | 2.712 | 2.670 | 2.648 | 2.612 | 2.588
|  UCL∗ | 0.136 | 0.203 | 0.266 | 0.325 | 0.381 | 0.435 | 0.488 | 0.539 | 0.588 | 0.636
| $L^{\ast\ast}$ | 2.842 | 2.786 | 2.751 | 2.705 | 2.674 | 2.647 | 2.617 | 2.590 | 2.560 | 2.536
|  UCL∗∗ | 0.122 | 0.193 | 0.258 | 0.318 | 0.375 | 0.431 | 0.485 | 0.537 | 0.587 | 0.636
15 | $L$ | 3.044 | 2.893 | 2.814 | 2.768 | 2.723 | 2.687 | 2.661 | 2.637 | 2.616 | 2.585
| UCL | 0.107 | 0.175 | 0.236 | 0.295 | 0.351 | 0.406 | 0.459 | 0.511 | 0.562 | 0.611
| $L^{\ast}$ | 3.001 | 2.873 | 2.806 | 2.761 | 2.720 | 2.687 | 2.657 | 2.639 | 2.619 | 2.587
|  UCL∗ | 0.120 | 0.185 | 0.245 | 0.302 | 0.357 | 0.410 | 0.462 | 0.513 | 0.563 | 0.611
| $L^{\ast\ast}$ | 2.762 | 2.740 | 2.706 | 2.682 | 2.652 | 2.623 | 2.604 | 2.582 | 2.565 | 2.537
|  UCL∗∗ | 0.107 | 0.175 | 0.236 | 0.295 | 0.352 | 0.406 | 0.459 | 0.511 | 0.562 | 0.611
20 | $L$ | 2.980 | 2.858 | 2.790 | 2.748 | 2.710 | 2.678 | 2.655 | 2.635 | 2.614 | 2.585
|  UCL1 | 0.098 | 0.164 | 0.224 | 0.282 | 0.337 | 0.391 | 0.444 | 0.496 | 0.547 | 0.596
| $L^{\ast}$ | 2.948 | 2.841 | 2.781 | 2.736 | 2.707 | 2.677 | 2.655 | 2.634 | 2.613 | 2.584
|  UCL∗ | 0.111 | 0.174 | 0.232 | 0.288 | 0.343 | 0.396 | 0.448 | 0.498 | 0.548 | 0.596
| $L^{\ast\ast}$ | 2.701 | 2.705 | 2.683 | 2.662 | 2.637 | 2.617 | 2.598 | 2.578 | 2.561 | 2.540
|  UCL∗∗ | 0.098 | 0.164 | 0.224 | 0.282 | 0.337 | 0.392 | 0.444 | 0.496 | 0.547 | 0.597

Table 5: Simulation results for ARL1 based on Table 1. |  | $p_{1}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
---|---|---|---|---|---|---|---|---|---|---|---|---
$\delta$ | $n$ | $p_{1}^{\ast}$ | 0.100 | 0.149 | 0.199 | 0.248 | 0.298 | 0.347 | 0.397 | 0.446 | 0.496 | 0.545
|  | $p_{1}^{\ast\ast}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
0.1 | 5 |  ARL1 | 208.0 | 166.6 | 136.4 | 119.8 | 103.0 | 89.1 | 80.0 | 70.2 | 61.0 | 53.6
|  |  ARL${}_{1}^{\ast}$ | 251.9 | 198.4 | 160.6 | 137.1 | 119.7 | 103.8 | 90.9 | 79.3 | 69.9 | 61.9
|  |  ARL${}_{1}^{\ast\ast}$ | 207.7 | 166.0 | 138.8 | 120.8 | 103.8 | 88.0 | 80.6 | 70.4 | 61.8 | 54.0
| 10 |  ARL1 | 168.3 | 123.6 | 101.7 | 82.4 | 69.7 | 60.0 | 51.2 | 44.8 | 37.8 | 33.4
|  |  ARL${}_{1}^{\ast}$ | 218.6 | 154.0 | 121.2 | 98.7 | 81.7 | 68.9 | 60.2 | 51.0 | 43.8 | 38.8
|  |  ARL${}_{1}^{\ast\ast}$ | 168.4 | 125.5 | 100.5 | 82.9 | 69.9 | 59.9 | 51.1 | 44.4 | 38.1 | 33.4
| 15 |  ARL1 | 146.3 | 102.4 | 80.4 | 65.6 | 54.5 | 45.5 | 38.9 | 33.4 | 28.0 | 24.5
|  |  ARL${}_{1}^{\ast}$ | 196.2 | 130.9 | 99.8 | 79.7 | 63.5 | 53.7 | 45.6 | 39.2 | 33.6 | 28.8
|  |  ARL${}_{1}^{\ast\ast}$ | 144.6 | 103.6 | 80.9 | 65.7 | 54.0 | 45.7 | 38.9 | 33.5 | 28.5 | 24.6
| 20 |  ARL1 | 128.9 | 88.8 | 67.6 | 53.6 | 44.8 | 37.1 | 31.5 | 26.9 | 22.9 | 19.8
|  |  ARL${}_{1}^{\ast}$ | 176.1 | 115.9 | 84.1 | 65.2 | 52.8 | 43.6 | 37.7 | 31.9 | 26.8 | 23.1
|  |  ARL${}_{1}^{\ast\ast}$ | 128.5 | 88.8 | 67.1 | 53.3 | 44.8 | 37.1 | 31.5 | 26.9 | 22.9 | 19.8
|  | $p_{1}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
|  | $p_{1}^{\ast}$ | 0.104 | 0.158 | 0.212 | 0.266 | 0.320 | 0.374 | 0.428 | 0.482 | 0.536 | 0.590
|  | $p_{1}^{\ast\ast}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
0.2 | 5 |  ARL1 | 128.7 | 89.2 | 66.7 | 54.8 | 44.9 | 36.6 | 32.3 | 27.1 | 23.1 | 19.7
|  |  ARL${}_{1}^{\ast}$ | 175.6 | 116.0 | 83.5 | 65.8 | 54.3 | 44.4 | 37.8 | 31.9 | 27.0 | 23.2
|  |  ARL${}_{1}^{\ast\ast}$ | 127.2 | 89.0 | 68.0 | 55.0 | 45.0 | 36.3 | 32.2 | 27.3 | 23.3 | 19.7
| 10 |  ARL1 | 91.8 | 58.4 | 43.9 | 33.2 | 27.3 | 22.8 | 18.8 | 16.2 | 13.4 | 11.8
|  |  ARL${}_{1}^{\ast}$ | 136.2 | 78.3 | 55.1 | 42.2 | 32.5 | 26.9 | 22.7 | 18.7 | 15.7 | 13.8
|  |  ARL${}_{1}^{\ast\ast}$ | 90.9 | 59.1 | 43.7 | 33.2 | 27.2 | 22.9 | 18.8 | 16.2 | 13.5 | 11.8
| 15 |  ARL1 | 73.0 | 44.9 | 32.5 | 25.1 | 20.2 | 16.6 | 14.0 | 12.0 | 9.8 | 8.6
|  |  ARL${}_{1}^{\ast}$ | 114.1 | 62.5 | 42.3 | 31.8 | 24.1 | 19.9 | 16.5 | 14.1 | 12.0 | 10.0
|  |  ARL${}_{1}^{\ast\ast}$ | 72.4 | 45.3 | 32.3 | 25.1 | 20.1 | 16.7 | 14.0 | 12.0 | 9.9 | 8.6
| 20 |  ARL1 | 61.3 | 37.1 | 26.1 | 19.9 | 16.4 | 13.2 | 11.2 | 9.4 | 8.1 | 6.8
|  |  ARL${}_{1}^{\ast}$ | 96.9 | 51.9 | 34.2 | 25.2 | 19.6 | 15.8 | 13.5 | 11.3 | 9.3 | 8.0
|  |  ARL${}_{1}^{\ast\ast}$ | 60.9 | 37.0 | 26.1 | 19.8 | 16.3 | 13.3 | 11.2 | 9.4 | 8.1 | 6.8

Table 6: Simulation results for ARL1 based on Table 2. |  | $p_{1}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
---|---|---|---|---|---|---|---|---|---|---|---|---
$\delta$ | $n$ | $p_{1}^{\ast}$ | 0.064 | 0.118 | 0.172 | 0.226 | 0.280 | 0.334 | 0.387 | 0.441 | 0.495 | 0.549
|  | $p_{1}^{\ast\ast}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
0.1 | 5 |  ARL1 | 209.4 | 165.7 | 137.2 | 119.5 | 103.7 | 89.1 | 80.3 | 69.7 | 61.5 | 54.2
|  |  ARL${}_{1}^{\ast}$ | 217.0 | 170.3 | 143.6 | 122.8 | 106.1 | 92.0 | 81.8 | 71.8 | 62.6 | 55.1
|  |  ARL${}_{1}^{\ast\ast}$ | 206.5 | 165.2 | 136.6 | 120.2 | 104.1 | 88.8 | 79.9 | 69.7 | 61.6 | 53.8
| 10 |  ARL1 | 169.3 | 124.5 | 101.4 | 82.8 | 69.9 | 60.3 | 51.0 | 44.8 | 37.9 | 33.3
|  |  ARL${}_{1}^{\ast}$ | 183.8 | 130.4 | 103.9 | 83.9 | 72.2 | 61.5 | 52.5 | 45.6 | 39.1 | 34.4
|  |  ARL${}_{1}^{\ast\ast}$ | 166.3 | 126.0 | 101.7 | 82.4 | 69.6 | 60.1 | 50.7 | 44.5 | 37.9 | 33.3
| 15 |  ARL1 | 146.2 | 102.5 | 80.9 | 65.3 | 54.4 | 45.6 | 38.8 | 33.5 | 28.1 | 24.5
|  |  ARL${}_{1}^{\ast}$ | 158.1 | 108.9 | 84.1 | 66.9 | 55.9 | 47.3 | 40.5 | 34.8 | 29.1 | 25.3
|  |  ARL${}_{1}^{\ast\ast}$ | 145.6 | 103.2 | 80.3 | 65.0 | 54.0 | 45.7 | 39.0 | 33.6 | 28.3 | 24.6
| 20 |  ARL1 | 130.0 | 89.2 | 67.3 | 53.4 | 44.7 | 37.1 | 31.6 | 26.7 | 22.9 | 19.6
|  |  ARL${}_{1}^{\ast}$ | 140.6 | 94.1 | 70.6 | 56.7 | 46.1 | 38.4 | 32.5 | 27.7 | 23.7 | 20.3
|  |  ARL${}_{1}^{\ast\ast}$ | 129.6 | 87.9 | 67.4 | 53.3 | 44.8 | 36.8 | 31.6 | 26.9 | 23.0 | 19.7
|  | $p_{1}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
|  | $p_{1}^{\ast}$ | 0.069 | 0.128 | 0.186 | 0.245 | 0.304 | 0.363 | 0.422 | 0.480 | 0.539 | 0.598
|  | $p_{1}^{\ast\ast}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
0.2 | 5 |  ARL1 | 128.2 | 89.0 | 66.7 | 54.7 | 45.0 | 36.6 | 32.2 | 27.2 | 23.1 | 19.8
|  |  ARL${}_{1}^{\ast}$ | 138.3 | 93.1 | 71.4 | 57.0 | 46.7 | 37.7 | 33.0 | 27.9 | 23.8 | 20.4
|  |  ARL${}_{1}^{\ast\ast}$ | 128.0 | 88.7 | 67.2 | 54.9 | 44.8 | 36.5 | 32.2 | 27.2 | 23.3 | 19.7
| 10 |  ARL1 | 92.0 | 58.6 | 43.8 | 33.3 | 27.2 | 22.8 | 18.8 | 16.2 | 13.4 | 11.8
|  |  ARL${}_{1}^{\ast}$ | 103.3 | 62.6 | 45.4 | 34.3 | 28.4 | 23.5 | 19.4 | 16.6 | 13.9 | 12.2
|  |  ARL${}_{1}^{\ast\ast}$ | 90.8 | 58.9 | 43.8 | 33.4 | 27.1 | 22.7 | 18.7 | 16.2 | 13.4 | 11.8
| 15 |  ARL1 | 72.8 | 44.9 | 32.4 | 25.0 | 20.2 | 16.7 | 14.0 | 12.0 | 9.8 | 8.6
|  |  ARL${}_{1}^{\ast}$ | 81.8 | 48.3 | 34.3 | 26.0 | 20.9 | 17.3 | 14.5 | 12.4 | 10.2 | 8.8
|  |  ARL${}_{1}^{\ast\ast}$ | 72.3 | 44.8 | 32.4 | 25.2 | 20.1 | 16.6 | 14.0 | 12.0 | 9.8 | 8.6
| 20 |  ARL1 | 61.1 | 37.0 | 26.2 | 19.9 | 16.3 | 13.2 | 11.2 | 9.5 | 8.0 | 6.8
|  |  ARL${}_{1}^{\ast}$ | 68.9 | 39.9 | 27.6 | 21.4 | 17.0 | 13.8 | 11.6 | 9.7 | 8.2 | 7.0
|  |  ARL${}_{1}^{\ast\ast}$ | 61.1 | 37.0 | 26.1 | 19.9 | 16.4 | 13.3 | 11.2 | 9.5 | 8.0 | 6.8

Table 7: Simulation results for ARL1 based on Table 3. |  | $p_{1}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
---|---|---|---|---|---|---|---|---|---|---|---|---
$\delta$ | $n$ | $p_{1}^{\ast}$ | 0.100 | 0.149 | 0.199 | 0.248 | 0.298 | 0.347 | 0.397 | 0.446 | 0.496 | 0.545
|  | $p_{1}^{\ast\ast}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
0.1 | 5 |  ARL1 | 242.9 | 209.1 | 184.0 | 166.6 | 148.2 | 132.7 | 119.0 | 108.3 | 98.0 | 88.5
|  |  ARL${}_{1}^{\ast}$ | 282.1 | 239.6 | 207.2 | 185.2 | 164.6 | 148.1 | 135.0 | 120.6 | 107.8 | 100.1
|  |  ARL${}_{1}^{\ast\ast}$ | 245.9 | 209.6 | 185.5 | 164.2 | 147.5 | 133.0 | 120.2 | 106.9 | 96.7 | 85.5
| 10 |  ARL1 | 213.2 | 171.9 | 142.9 | 124.2 | 107.0 | 92.9 | 81.6 | 70.4 | 62.0 | 53.0
|  |  ARL${}_{1}^{\ast}$ | 257.3 | 203.4 | 166.7 | 142.2 | 124.2 | 107.9 | 93.9 | 82.6 | 71.6 | 61.8
|  |  ARL${}_{1}^{\ast\ast}$ | 213.3 | 172.0 | 144.6 | 124.5 | 107.3 | 93.0 | 81.1 | 70.9 | 61.6 | 53.1
| 15 |  ARL1 | 190.4 | 147.4 | 120.9 | 101.6 | 85.0 | 73.0 | 62.5 | 53.3 | 45.3 | 38.6
|  |  ARL${}_{1}^{\ast}$ | 232.8 | 178.9 | 142.8 | 118.6 | 99.3 | 85.6 | 72.7 | 62.8 | 53.8 | 45.6
|  |  ARL${}_{1}^{\ast\ast}$ | 190.1 | 147.9 | 120.3 | 100.3 | 85.3 | 72.8 | 62.5 | 53.3 | 45.2 | 38.6
| 20 |  ARL1 | 175.0 | 130.6 | 104.2 | 85.1 | 71.5 | 59.3 | 50.3 | 42.8 | 36.2 | 29.9
|  |  ARL${}_{1}^{\ast}$ | 223.3 | 161.8 | 126.4 | 102.5 | 83.2 | 70.4 | 60.2 | 50.3 | 42.8 | 35.7
|  |  ARL${}_{1}^{\ast\ast}$ | 174.1 | 130.7 | 104.4 | 85.1 | 71.0 | 59.7 | 50.2 | 42.8 | 35.8 | 30.4
|  | $p_{1}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
|  | $p_{1}^{\ast}$ | 0.104 | 0.158 | 0.212 | 0.266 | 0.32 | 0.374 | 0.428 | 0.482 | 0.536 | 0.59
|  | $p_{1}^{\ast\ast}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
0.2 | 5 |  ARL1 | 166.8 | 128.4 | 102.4 | 85.4 | 70.6 | 59.6 | 50.4 | 43.1 | 36.7 | 31.1
|  |  ARL${}_{1}^{\ast}$ | 217.7 | 159.7 | 124.5 | 101.8 | 84.2 | 70.7 | 60.3 | 50.5 | 42.7 | 37.1
|  |  ARL${}_{1}^{\ast\ast}$ | 169.7 | 128.6 | 104.3 | 84.3 | 70.2 | 59.6 | 50.4 | 42.4 | 36.3 | 30.1
| 10 |  ARL1 | 132.2 | 91.4 | 68.0 | 53.7 | 43.0 | 34.8 | 29.0 | 23.9 | 20.0 | 16.6
|  |  ARL${}_{1}^{\ast}$ | 182.1 | 119.2 | 86.6 | 66.2 | 52.9 | 43.0 | 34.9 | 28.9 | 23.9 | 19.9
|  |  ARL${}_{1}^{\ast\ast}$ | 131.1 | 90.5 | 68.2 | 53.8 | 42.9 | 34.9 | 28.9 | 23.9 | 19.9 | 16.7
| 15 |  ARL1 | 108.9 | 70.9 | 51.8 | 39.5 | 31.0 | 25.1 | 20.5 | 16.9 | 13.8 | 11.5
|  |  ARL${}_{1}^{\ast}$ | 156.0 | 96.4 | 66.7 | 49.6 | 38.1 | 30.7 | 24.8 | 20.4 | 16.7 | 13.9
|  |  ARL${}_{1}^{\ast\ast}$ | 108.9 | 70.7 | 51.7 | 39.2 | 30.9 | 25.1 | 20.4 | 16.8 | 13.8 | 11.5
| 20 |  ARL1 | 93.4 | 58.1 | 41.3 | 30.8 | 24.4 | 19.4 | 15.7 | 13.0 | 10.7 | 8.9
|  |  ARL${}_{1}^{\ast}$ | 141.0 | 81.4 | 55.1 | 40.0 | 30.0 | 23.8 | 19.4 | 15.6 | 12.9 | 10.7
|  |  ARL${}_{1}^{\ast\ast}$ | 93.6 | 58.1 | 41.3 | 30.9 | 24.4 | 19.4 | 15.8 | 13.1 | 10.8 | 9.0

Table 8: Simulation results for ARL1 based on Table 4. |  | $p_{1}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
---|---|---|---|---|---|---|---|---|---|---|---|---
$\delta$ | $n$ | $p_{1}^{\ast}$ | 0.064 | 0.118 | 0.172 | 0.226 | 0.280 | 0.334 | 0.387 | 0.441 | 0.495 | 0.549
|  | $p_{1}^{\ast\ast}$ | 0.055 | 0.110 | 0.165 | 0.220 | 0.275 | 0.330 | 0.385 | 0.440 | 0.495 | 0.550
| 5 |  ARL1 | 242.0 | 208.2 | 184.5 | 166.5 | 147.2 | 131.9 | 118.8 | 109.1 | 98.7 | 87.8
|  |  ARL${}_{1}^{\ast}$ | 258.2 | 217.8 | 190.0 | 167.5 | 151.6 | 138.4 | 122.5 | 109.2 | 97.9 | 90.4
|  |  ARL${}_{1}^{\ast\ast}$ | 244.2 | 211.1 | 185.5 | 164.7 | 147.0 | 133.1 | 120.0 | 107.4 | 96.0 | 85.5
| 10 |  ARL1 | 214.5 | 172.2 | 142.7 | 124.0 | 106.6 | 93.4 | 81.9 | 70.7 | 62.1 | 53.0
|  |  ARL${}_{1}^{\ast}$ | 222.0 | 175.9 | 147.9 | 128.0 | 110.7 | 95.9 | 83.6 | 73.4 | 63.4 | 55.0
|  |  ARL${}_{1}^{\ast\ast}$ | 213.4 | 171.0 | 143.8 | 123.2 | 107.5 | 93.0 | 81.6 | 71.1 | 61.6 | 53.3
| 15 |  ARL1 | 190.0 | 146.8 | 120.6 | 101.3 | 85.4 | 72.8 | 62.1 | 53.2 | 45.2 | 38.2
|  |  ARL${}_{1}^{\ast}$ | 201.4 | 154.7 | 125.5 | 104.5 | 87.7 | 75.2 | 64.1 | 55.3 | 46.8 | 39.7
|  |  ARL${}_{1}^{\ast\ast}$ | 191.0 | 147.9 | 120.4 | 101.3 | 85.2 | 73.0 | 62.7 | 53.2 | 45.4 | 38.5
| 20 |  ARL1 | 173.9 | 131.1 | 103.7 | 85.6 | 71.4 | 59.4 | 50.1 | 42.6 | 36.0 | 30.1
|  |  ARL${}_{1}^{\ast}$ | 185.6 | 137.6 | 108.2 | 89.1 | 74.4 | 61.7 | 52.5 | 44.2 | 37.3 | 31.1
|  |  ARL${}_{1}^{\ast\ast}$ | 173.1 | 130.3 | 104.4 | 85.1 | 70.7 | 59.6 | 50.3 | 42.7 | 35.9 | 30.2
|  | $p_{1}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
|  | $p_{1}^{\ast}$ | 0.069 | 0.128 | 0.186 | 0.245 | 0.304 | 0.363 | 0.422 | 0.480 | 0.539 | 0.598
|  | $p_{1}^{\ast\ast}$ | 0.060 | 0.120 | 0.180 | 0.240 | 0.300 | 0.360 | 0.420 | 0.480 | 0.540 | 0.600
0.2 | 5 |  ARL1 | 166.6 | 129.0 | 103.1 | 85.7 | 70.2 | 59.4 | 50.0 | 43.0 | 36.7 | 31.0
|  |  ARL${}_{1}^{\ast}$ | 183.6 | 135.5 | 108.4 | 87.4 | 72.8 | 62.5 | 52.2 | 43.8 | 37.3 | 32.0
|  |  ARL${}_{1}^{\ast\ast}$ | 169.0 | 129.1 | 102.7 | 84.8 | 70.3 | 59.6 | 50.3 | 42.5 | 36.0 | 30.2
| 10 |  ARL1 | 132.4 | 91.1 | 67.6 | 54.0 | 43.0 | 34.9 | 29.0 | 23.8 | 20.0 | 16.6
|  |  ARL${}_{1}^{\ast}$ | 141.0 | 95.2 | 70.9 | 55.7 | 44.7 | 36.1 | 30.1 | 25.0 | 20.7 | 17.2
|  |  ARL${}_{1}^{\ast\ast}$ | 129.9 | 90.5 | 68.0 | 53.8 | 43.2 | 34.7 | 29.0 | 23.9 | 20.0 | 16.5
| 15 |  ARL1 | 108.7 | 70.6 | 51.6 | 39.5 | 31.1 | 24.9 | 20.5 | 16.9 | 13.8 | 11.5
|  |  ARL${}_{1}^{\ast}$ | 119.9 | 76.1 | 54.2 | 41.3 | 32.5 | 26.1 | 21.1 | 17.6 | 14.4 | 11.9
|  |  ARL${}_{1}^{\ast\ast}$ | 109.5 | 71.0 | 51.7 | 39.5 | 31.0 | 25.0 | 20.5 | 16.9 | 13.8 | 11.6
| 20 |  ARL1 | 92.8 | 58.1 | 41.4 | 31.0 | 24.3 | 19.3 | 15.7 | 13.0 | 10.7 | 8.9
|  |  ARL${}_{1}^{\ast}$ | 104.3 | 62.9 | 43.6 | 32.9 | 25.6 | 20.2 | 16.4 | 13.5 | 11.2 | 9.3
|  |  ARL${}_{1}^{\ast\ast}$ | 92.8 | 57.9 | 41.3 | 30.9 | 24.5 | 19.3 | 15.8 | 13.0 | 10.8 | 9.0
Table 9: Numerical results based on naive estimator. | Naive |  |  Correct ($\pi=0.95$) |  |  Correct ($\pi=0.99$)
---|---|---|---|---|---
$\lambda$ | $L^{\ast}$ |  UCL∗ |  | $L^{\ast\ast}$ |  UCL∗∗ |  | $L^{\ast\ast}$ |  UCL∗∗
0.05 | 2.222 | 0.126 |  | 2.013 | 0.080 |  | 2.183 | 0.118
0.20 | 2.753 | 0.151 |  | 2.019 | 0.101 |  | 2.616 | 0.142

---
Figure 1: The naive and corrected EWMA $p$-control charts under $\lambda=0.05$
and IC samples.
---
Figure 2: The naive and corrected EWMA $p$-control charts under $\lambda=0.20$
and IC samples.
---
Figure 3: The naive and corrected EWMA $p$-control charts under $\lambda=0.05$
and OC samples. Red points are OC detection.
---
Figure 4: The naive and corrected EWMA $p$-control charts under $\lambda=0.20$
and OC samples. Red points are OC detection.

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