The Unconditional Performance of Control Charts for Zero-Inflated Processes
with Estimated Parameters

1Athanasios C. RAKITZIS ***Corresponding author. E-mail address:
<EMAIL_ADDRESS>
This is an original manuscript of an article that will be published by Taylor
& Francis in Statistical Methods and Applications in Systems Assurance &
Quality of the Book Series Advanced Research in Reliability and System
Assurance, 2Eftychia MAMZERIDOU, 3Petros E. MARAVELAKIS

1Department of Statistics and Insurance Science, University of Piraeus,
Piraeus 18534, Greece.

2Department of Statistics and Actuarial-Financial Mathematics, University of
the Aegean, Karlovasi 83200, Samos, Greece.

3Department of Business Administration, University of Piraeus, Piraeus 18534,
Greece.

###### Abstract

Control charts for zero-inflated processes have attracted the interest of the
researchers in the recent years. In this work we investigate the performance
of Shewhart-type charts for zero-inflated Poisson and zero-inflated Binomial
processes, in the case of estimated parameters. This is a case that usually
occurs in practice, especially prior to starting the process monitoring. Using
Monte Carlo simulation we evaluate charts’ performance under an unconditional
perspective and provide guidelines for their use in practice. We examine both
the in-control and the out-of-control performance.

Keywords: Moment estimators, Maximum likelihood estimators, Unconditional run
length distribution, zero-inflated binomial distribution, zero-inflated
Poisson distribution.

##  1 Introduction

Control charts are the main tool of Statistical Process Control (SPC) and can
be used for monitoring a process in order to detect changes in it. Walter A.
Shewhart (Shewhart (1926)) was the first who proposed the use of control
charts, back in the 1920’s. The most common and often used are the Shewhart-
type charts because of their ease in design and interpretation. Two main
categories of control charts are the variables charts and the attributes
charts. Control charts for variables are used when the quality characteristic
$X$ (random variable, r.v.) follows a continuous distribution while if $X$
follows a discrete distribution then the respective chart is a control chart
for attributes. Control charts for attributes are used for process monitoring
when the data collected by the process are counts. For example, attributes
control charts are used for monitoring the number of non-conforming items in a
sample, the number of non-conformities in each inspected unit, the number of
new daily infections in a specific region or the daily number of logins in the
computers of a workstation.

The $np$\- and $c$-chart are two common Shewhart-type attributes charts. The
first is used when $X$ is a binomial random variable (such as the number of
non-conforming items in the sample) while the second is used when $X$ is a
Poisson random variable (such as the number of emergency calls in a hospital
during the night shift). See Montgomery (2009) for the basic properties of the
$np$\- and $c$-chart.

However, there are several processes where the $np$\- and $c$-chart cannot be
used for their monitoring because the distributional assumption of the
binomial and the Poisson distribution is violated. For example, when there is
overdispersion in the data or if there is an excessive number of zero values
(or, sometimes, both). Therefore, practitioners have to select first the model
that fits better their data and then develop a control chart based on this
model.

Zero-inflated models (see Johnson et al. (2005), pp. 351-356) have been
recommended as alternative models that take into account the excessive number
of zeros as well as the overdispersion that is caused by them. This excess in
zeros is attributed to the technological progress and automation of
manufacturing technology which led to high-quality (or high-yield) processes;
processes with a very low probability of non-conforming items or with a very
low rate in the occurrence of non-conforming items. A zero value occurs when
there are no non-conforming items in a sample or no non-conformities in the
inspected unit.

In recent years, there has been an increased interest in control charts for
monitoring zero-inflated processes. See the recent overview by Mahmood and Xie
(2019). However, the majority of these works is about control charts for Phase
II monitoring, where the basic assumption is the values of the process
parameters are known. This is a strong assumption which rarely (if ever) is
met in practice. According to Montgomery (2009) (see also Jensen et al.
(2006)) there is significant difference between the theoretical and the actual
performance of a control chart, which is attributed to the presence of
estimation error. With the term theoretical we refer to the case where process
parameters are known (or Case K) while the term actual performance refers to
the case where the process parameters are unknown and must be estimated (Case
U). Albers et al. Albers et al. (2004) discriminated the estimation error into
two distinct sources, namely model error and stochastic error. The first is
caused by incorrect model assumptions (e.g. the true distributional model is
much different than the selected one) while the latter is due to estimation
uncertainty and is related to the selected Phase I sample. In this work the
focus is solely on the stochastic (estimation) error. He et al. He et al.
(2003) where the first who studied the estimation effect on the performance of
a Shewhart-type control chart for monitoring a zero-inflated Poisson process.
Rakitzis and Castagliola (2016) extended the work of He et al. (2003) and
studied also the performance of a Shewhart-type chart for monitoring a zero-
inflated binomial process. Both works assume that only one of the two
parameters of these zero-inflated processes is unknown. Recently, Mamzeridou
and Rakitzis (2023) investigated the performance of an upper one-sided
Shewhart chart for monitoring a zero-inflated Poisson process, in the case
where both process parameters are unknown.

In this work we consider two popular zero-inflated models (namely, the zero-
inflated Poisson and the zero-inflated binomial distribution) and study the
statistical design and performance of the corresponding Shewhart-type charts
in Case U. It should be mentioned that control charts for monitoring a zero-
inflated binomial process in Case U have not been considered so far. In
addition, we consider two estimation methods, maximum likelihood estimation
(MLE) and method of moments (MoM), while the proposed chart is two-sided with
Shewhart-type limits, instead of probability limits. The main difference with
the work of Mamzeridou and Rakitzis (2023) is that in this work we evaluate
the performance of the charts in Case U from a different perspective. Thus,
different performance measures are used. This will be explained later.

The structure of this paper is: In Section 2 we provide, in brief, the basic
properties of the Shewhart charts for monitoring a zero-inflated Poisson
(Section 2.1) and a zero-inflated binomial (Section 2.2) process, in the case
of known parameters (Case K), along with the appropriate measures for their
performance (Section 2.3). In Section 3, we describe the implementation of the
charts in Case U, for each of the considered processes, providing also the
estimators under each case. Using these estimates we derive the basic
properties of the considered Shewhart charts in the case of estimated
parameters. Also, we present the steps for designing the charts as well as the
appropriate performance measures, under an unconditional perspective. In
Section 4 we present the findings of a simulation study regarding the in-
control (IC) and the out-of-control (OOC) performance of the considered charts
in Case U. The results in the case of monitoring a ZIP process are given in
Section 4.1 while in Section 4.2 we provide the results in the case of a ZIB
process. A brief discussion on the OOC performance of the charts is given in
Section 5 while conclusions are summarized in Section 6.

##  2 Shewhart Charts for Monitoring ZIP and ZIB Processes

###  2.1 The Shewhart Chart for ZIP Processes (Case K)

The zero-inflated Poisson (ZIP) distribution is a generalization of the
Poisson distribution that can be used to model count data containing an
excessive number of zeros. By definition, if $X$ is a ZIP random variable, it
is defined on $\left\\{0,1,\ldots\right\\}$ (as for the standard Poisson
distribution) and its probability mass function (p.m.f.) is given by

$f_{ZIP}\left(x\right.\left|\phi,\lambda\right)=\left\\{\begin{array}[]{rcl}\phi+\left(1-\phi\right)f_{P}\left(0\right.\left|\lambda\right),&x=&0\\\
\left(1-\phi\right)f_{P}\left(x\right.\left|\lambda\right),&x=&1,2,...\end{array}\right.,$
(1)

where $f_{P}\left(x\right.\left|\lambda\right)$ is the p.m.f. of the standard
Poisson distribution with parameter $\lambda>0$ and $\phi\in\left[0,1\right]$
is the zero-inflation parameter. If $\phi=0$, then the ZIP distribution
coincides with the standard Poisson distribution. If $\phi=1$, then the ZIP
distribution reduces to the Dirac distribution on $x=0$. Moreover, it is not
difficult to verify that the cumulative distribution function (c.d.f.) of $X$
is given by:

$F_{ZIP}\left(x\right.\left|\phi,\lambda\right)=\phi+(1-\phi)F_{P}\left(x\right.\left|\lambda\right),$
(2)

where $F_{P}\left(x\right.\left|\lambda\right)$ is the c.d.f. of the Poisson
distribution with parameter $\lambda$. The mean and the variance of the ZIP
distribution with parameters $\left(\phi,\lambda\right)$ are, respectively,
given by

$E(X)=\lambda\left(1-\phi\right),\;\;V(X)=\lambda\left(1+\lambda\phi\right)\left(1-\phi\right).$
(3)

Let us now assume that we want to set-up a Shewhart control chart for
monitoring a ZIP process. In the case of known parameters (Case K), the upper
and the lower control limits (denoted as $UCL_{ZIP}$, $LCL_{ZIP}$,
respectively) are given by (see, for example, Rakitzis and Castagliola (2016),
page 4196):

$UCL_{ZIP}=\left\lfloor\lambda_{0}(1-\phi_{0})+L\sqrt{\lambda_{0}(1+\lambda_{0}\phi_{0})(1-\phi_{0})}\right\rfloor,$
(4)
$LCL_{ZIP}=\max\left(0,\left\lceil\lambda_{0}(1-\phi_{0})-L\sqrt{\lambda_{0}(1+\lambda_{0}\phi_{0})(1-\phi_{0})}\right\rceil\right),$
(5)

where $\left\lfloor\ldots\right\rfloor$ and $\left\lceil\ldots\right\rceil$
denote the rounded down and rounded up integer, $\phi_{0}$ $\lambda_{0}$ are
the in-control values of the process parameters $\phi$ and $\lambda$, while
$L>0$ is a constant that plays the role of chart’s design parameter. These
limits are also known as Shewhart-type or $L\sigma$ limits. It should be noted
that in Mamzeridou and Rakitzis (2023) the authors used only one probability
limit and not Shewhart-type limits. Also, given $\phi_{0}$ and $\lambda_{0}$,
the value of $L$ determines the limits of the interval
$[LCL_{ZIP},UCL_{ZIP}]$. Thus, it is directly related to the false alarm rate
(FAR) of the chart. Recall that the FAR is the probability for the chart to
give an OOC signal when the process is actually IC. The largest the $L$ the
smallest the FAR.

Let $Y_{1},Y_{2},\ldots$ be independent random variables such that $Y_{i}\sim
ZIP(\phi_{1},\lambda_{1})$, i.e. a ZIP distribution with parameters
$(\phi_{1},\lambda_{1})$, where, in general, $\phi_{1}\neq\phi_{0}$,
$\lambda_{1}\neq\lambda_{0}$ and $\phi_{1}\in[0,1]$, $\lambda_{1}>0$. If
$\phi_{1}=\phi_{0}$ and $\lambda_{1}=\lambda_{0}$, then the process is IC,
otherwise it is OOC. Then, the probability $\beta$ which is the probability
that the number $Y_{i}$ of non-conformities is within the interval
$[LCL_{ZIP},UCL_{ZIP}]$ equals

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle P\left(LCL_{ZIP}\leq
Y_{i}\leq UCL_{ZIP}\right)$ (6) $\displaystyle=$ $\displaystyle
F_{ZIP}\left(\left.UCL_{ZIP}\right|\phi_{1},\lambda_{1}\right)-F_{ZIP}\left(\left.LCL_{ZIP}-1\right|\phi_{1},\lambda_{1}\right)$

###  2.2 The Shewhart Chart for ZIB Processes (Case K)

The zero-inflated binomial (ZIB) distribution is a generalization of the
binomial distribution that takes into account the excessive number of zeros,
compared to the usual binomial distribution. By definition, if $X$ is a ZIB
random variable, it is defined on $\left\\{0,1,\ldots,n\right\\}$ (same as the
binomial distribution) and its p.m.f. is given by

$f_{ZIB}\left(x\right.\left|\phi,n,p\right)=\left\\{\begin{array}[]{rcl}\phi+\left(1-\phi\right)f_{B}\left(0\right.\left|n,p\right),&x=&0\\\
\left(1-\phi\right)f_{B}\left(x\right.\left|n,p\right),&x=&1,2,...,n\end{array}\right.,$
(7)

where $f_{B}\left(x\right.\left|n,p\right)$ is the p.m.f. of the binomial
distribution $B(n,p)$ with parameters $n\in\\{1,2,\ldots\\}$ and $p\in(0,1)$.
Parameter $p$ is the success probability (i.e. the probability of an item
being non-conforming) while $\phi$ is the zero-inflation parameter. If
$\phi=0$, the ZIB distribution coincides with the binomial distribution with
parameters $n$ and $p$ while, for $\phi=1$, the ZIB distribution reduces to
the Dirac distribution on $x=0$. The c.d.f. of $X$ is given by

$F_{ZIB}\left(x\right.\left|\phi,n,p\right)=\phi+(1-\phi)F_{B}\left(x\right.\left|n,p\right),$
(8)

where $F_{B}\left(x\right.\left|n,p\right)$ is the c.d.f. of $B(n,p)$. The
mean and the variance of the ZIB distribution with parameters
$\left(\phi,n,p\right)$ are, respectively, given by the following two
expressions

$E(X)=np\left(1-\phi\right),\;\;V(X)=np\left(1-p+np\phi\right)\left(1-\phi\right).$
(9)

In a similar manner, the control limits of a the Shewhart chart for monitoring
a ZIB process with known parameters are given by (see, for example, Rakitzis
and Castagliola (2016), page 4196):

$UCL_{ZIB}=\left\lfloor
np_{0}(1-\phi_{0})+L\sqrt{np_{0}(1-p_{0}+np_{0}\phi_{0})(1-\phi_{0})}\right\rfloor,$
(10) $LCL_{ZIB}=\max\left(0,\left\lceil
np_{0}(1-\phi_{0})-L\sqrt{np_{0}(1-p_{0}+np_{0}\phi_{0})(1-\phi_{0})}\right\rceil\right),$
(11)

where $\phi_{0}$, $p_{0}$ are the in-control values of process parameters
$\phi$ and $p$, while constant $L>0$ is the chart’s design parameter. Let
$Y_{1},Y_{2},\ldots$ be independent random variables with $Y_{i}\sim
ZIB(\phi_{1},n,p_{1})$, i.e., the number of non-conforming units in a sample
of size $n$ is a ZIB random variable with parameters $(\phi_{1},n,p_{1})$.
Again, in general, $\phi_{1}\neq\phi_{0}$ and $p_{1}\neq p_{0}$ with
$\phi_{1}\in[0,1]$ and $p_{1}\in(0,1)$. If $\phi_{1}=\phi_{0}$ and
$p_{1}=p_{0}$ the process is IC, otherwise it is OOC. Then, the probability
$\beta$ which is the probability that the number $Y_{i}$ of non-conforming
items is within the interval $[LCL_{ZIB},UCL_{ZIB}]$ equals

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle P\left(LCL_{ZIB}\leq
Y_{i}\leq UCL_{ZIB}\right)$ (12) $\displaystyle=$ $\displaystyle
F_{ZIB}\left(\left.UCL_{ZIP}\right|\phi_{1},n,p_{1}\right)-F_{ZIP}\left(\left.LCL_{ZIP}-1\right|\phi_{1},n,p_{1}\right)$

###  2.3 Run length properties (Case K)

The run length of the Shewhart charts for ZIP and ZIB processes with known
parameters is a geometric random variable $N$ with parameter $1-\beta$, i.e.,
the p.m.f. $f_{N}(\nu)$ and the c.d.f. $F_{N}(\nu)$ of $N$ are defined for
$\nu=1,2,\ldots$ and they are, respectively, given by:

$f_{N}(\nu)=(1-\beta)\beta^{\nu-1},\;\;F_{N}(\nu)=1-\beta^{\nu},$ (13)

where $\beta$ is the probability defined in equations (6) and (12). Therefore,
the Average Run Length ($ARL$) and the Standard Deviation Run Length ($SDRL$)
of the Shewhart charts for ZIP and ZIB processes with known parameters equal

$E(N)=ARL=\frac{1}{1-\beta},\>\>\sqrt{V(N)}=SDRL=\frac{\sqrt{\beta}}{1-\beta}.$
(14)

Note also that the moment about the origin of order two of the run length
distribution equals

$\mu^{\prime}_{2}=E(N^{2})=V(N)+E^{2}(N)=\frac{\beta}{(1-\beta)^{2}}+\frac{1}{(1-\beta)^{2}}=\frac{1+\beta}{(1-\beta)^{2}},$
(15)

where $\mu^{\prime}_{k}=E(N^{k})$, $k=1,2,\ldots$. It should be mentioned that
since $\beta$ is given by equations (6) or (12) the run length distribution as
well as the $ARL$ and $SDRL$ refer all to the OOC case. For
$\phi_{1}=\phi_{0}$ and $\lambda_{1}=\lambda_{0}$ we get the respective IC
measures.

##  3 Shewhart Charts for Monitoring ZIP and ZIB Processes With Unknown
Parameters

In this section we discuss the properties of the two Shewhart charts, for ZIP
and ZIB processes, in Case U. The two processes are treated individually. As
already stated, for estimating the process parameters, we consider both the
MLE method and the MoM.

###  3.1 The Shewhart Chart for ZIP Processes (Case U)

Suppose that we are interested in monitoring a ZIP process but the IC values
of the process parameters are unknown. Thus, $\phi_{0}$ and $\lambda_{0}$ must
be estimated from a Phase I sample. Let $X_{1},X_{2},\ldots,X_{m}$ be a random
sample from $ZIP(\phi_{0},\lambda_{0})$, i.e., $X_{i}\sim
ZIP(\phi_{0},\lambda_{0})$, $i=1,2,\ldots,m$. This is the Phase I sample which
is used for estimating $\phi_{0}$ and $\lambda_{0}$.

Using the MoM, the respective moment estimators (ME) for $\lambda_{0}$ and
$\phi_{0}$ are given by:

$\tilde{\lambda}=\frac{\overline{X^{2}}}{\bar{X}}-1,\;\tilde{\phi}=1-\bar{X}/\tilde{\lambda}_{0},$
(16)

where

$\bar{X}=\frac{1}{m}\sum_{i=1}^{m}X_{i},\;\;\overline{X^{2}}=\frac{1}{m}\sum_{i=1}^{m}X_{i}^{2},$
(17)

Also, using the MLE method, the respective estimators are not given in closed
form but they are obtained numerically by solving the following non-linear
system of equations:

$\left\\{\begin{array}[]{l}\hat{\lambda}=\bar{X}^{+}(1-e^{-\hat{\lambda}})\\\
\hat{\phi}=1-\bar{X}/\hat{\lambda}\end{array}\right.$ (18)

where $\bar{X}^{+}$ is the average of strictly positive $X$’s.

When $\phi_{0}$, $\lambda_{0}$ are estimated by $\hat{\phi}$, $\hat{\lambda}$,
the control limits of the Shewhart chart are denoted as $\widehat{LCL}_{ZIP}$,
$\widehat{UCL}_{ZIP}$ and they are simply obtained by substituting $\phi_{0}$,
$\lambda_{0}$ with $\hat{\phi}$ and $\hat{\lambda}$ in equations (5) and (4).
In a similar manner, if $\phi_{0}$ and $\lambda_{0}$ are estimated by
$\tilde{\phi}$, $\tilde{\lambda}$, the respective limits are denoted as
$\widetilde{LCL}_{ZIP}$, $\widetilde{UCL}_{ZIP}$. It is not difficult to see
that in Case U the control limits are based on the estimates of process
parameters obtained from the available Phase I sample and thus, they are
random variables. Given the estimates of $\phi_{0}$ and $\lambda_{0}$ it is
possible to calculate the limits and proceed to Phase II process monitoring.

It is not difficult to realize that different practitioners will end up with
different estimates for the process parameters, because of the different Phase
I samples. Thus, it is possible to use different control limits. In order to
reflect the estimation error that it is attributed to the different estimates
among all the practitioners, we will use the probability
$\hat{\beta}=P(Y_{i}\in[\widehat{LCL}_{ZIP},\widehat{UCL}_{ZIP}]\left|\phi_{1},\lambda_{1}\right.)$
(or
$\tilde{\beta}=P(Y_{i}\in[\widetilde{LCL}_{ZIP},\widetilde{UCL}_{ZIP}]\left|\phi_{1},\lambda_{1}\right.)$,
in case of MoM estimators) to investigate the performance of the ZIP-Shewhart
chart in Case U. Note also that this probability cannot be calculated in
practice because $\phi_{1}$, $\lambda_{1}$ is unknown.

The assessment of the control chart’s performance in Case U is made under two
perspectives, the ‘unconditional’ or the ‘conditional’. In Case U, the run
length distribution of the ZIP-Shewhart chart is no longer a geometric one
because the control limits are r.v. However, given the estimates $\hat{\phi}$,
$\hat{\lambda}$ (or $\tilde{\phi}$, $\tilde{\lambda}$), the run length of the
chart becomes conditional on them and thus, it is known conditional run
length. Its distribution is now a geometric one. Using the conditional run
length distribution it is possible to evaluate the performance of the chart
under the ‘conditional’ perspective. This approach was followed by Mamzeridou
and Rakitzis (2023). See also Zhao and Driscoll (2016); Vakilian et al. (2018)
regarding the use of the conditional run length distribution to study the
$c$-chart in Case U.

On the other hand, by averaging the conditional run length distribution over
the (joint) distribution of $(\hat{\phi},\hat{\lambda})$ (or
$(\tilde{\phi},\tilde{\lambda})$) we get the unconditional run length
distribution. Using the unconditional run length distribution, the performance
of the chart is evaluated under the unconditional perspective. This approach
was followed by He et al. (2003) and Rakitzis and Castagliola (2016). The
majority of the literature on the performance of attributes control charts in
Case U is based on the use of the ‘unconditional’ run length distribution
(see, for example, Braun (1999); Testik et al. (2006); Castagliola and Wu
(2012); Castagliola et al. (2014); Wu et al. (2016); Johannssen et al. (2022)
and references therein). An advantage of this perspective is that it can be
applied before collecting the Phase I sample. Thus, it is possible to have
practical guidelines regarding the size of the Phase I samples that gives the
desired IC performance. For further details on the use of the unconditional
run length distribution in studying the performance of control charts in Case
U, see Jensen et al. (2006). Motivated by the previously mentioned works, we
will use the unconditional (or marginal) run length distribution.

In Algorithm 1 we provide the steps for the calculation of the appropriate
performance measures of the ZIP-Shewhart chart in Case U. To the best of our
knowledge, the joint distribution of $(\hat{\phi},\hat{\lambda})$ (or
$(\tilde{\phi},\tilde{\lambda})$) is not known. For this reason, we use Monte
Carlo simulation. The steps are given in the case of MLE method while in the
case of the MoM, the steps are similar. In order to avoid additional
complexities in notation we will use the notations $ARL$, $\mu^{\prime}_{2}$
and $SDRL$ in either Case K and Case U. However, in the latter case, these are
the unconditional measures.

Algorithm 1 Calculation of the unconditional performance measures - ZIP
process

1:Choose a Phase I sample of size $m$ from a $ZIP(\phi_{0},\lambda_{0})$
process.

2:Use the Phase I sample and estimate $\phi_{0}$, $\lambda_{0}$ by using the
MLE method (and get the $\hat{\phi}$, $\hat{\lambda}$ estimates).

3:Given the values $L$, $\hat{\phi}$ and $\hat{\lambda}$, determine
$\widehat{UCL}_{ZIP}$, $\widehat{LCL}_{ZIP}$ from equations (4) and (5),
respectively.

4:Calculate the IC $ARL$ and the IC $\mu^{\prime}_{2}$ as
$1/\left(1-\hat{\beta}\right)$ and
$\left(1+\hat{\beta}\right)/\left(1-\hat{\beta}\right)^{2}$, respectively.

5:Repeat Steps 1-4 $T$ times and calculate (i) the unconditional IC $ARL$ as
the average of the $T$ values $1/\left(1-\hat{\beta}\right)$, (ii) the
unconditional IC $\mu^{\prime}_{2}$ as the average of the $T$ values
$\left(1+\hat{\beta}\right)/\left(1-\hat{\beta}\right)^{2}$, (iii) the
unconditional IC $SDRL$ as $\sqrt{\mu^{\prime}_{2}-ARL^{2}}$, where $ARL$ and
$\mu^{\prime}_{2}$ are those obtained in (i) and (ii).

###  3.2 The Shewhart Chart for ZIB Processes (Case U)

Let $X_{1},X_{2},\ldots,X_{m}$ be a random sample from the zero-inflated
binomial (ZIB) distribution with in-control parameters $n$, $p_{0}$ and
$\phi_{0}$, i.e., $X_{i}\sim ZIB(\phi_{0},n,p_{0})$, $i=1,2,\ldots,m$. Using
the MoM, the respective estimates of the IC process parameters are given by:

$\tilde{p}=\frac{\overline{X^{2}}-\bar{X}}{(n-1)\bar{X}},\;\tilde{\phi}=1-\frac{(n-1)\bar{X}^{2}}{n\left(\overline{X^{2}}-\bar{X}\right)},$
(19)

where

$\bar{X}=\frac{1}{m}\sum_{i=1}^{m}X_{i},\;\;\overline{X^{2}}=\frac{1}{m}\sum_{i=1}^{m}X_{i}^{2},$
(20)

Also, similar to the case of ZIP process, the MLEs are not given in closed
form but they are obtained numerically by solving the following non-linear
system of equations:

$\left\\{\begin{array}[]{l}\hat{p}=\bar{X}^{+}(1-(1-\hat{p})^{n})\\\
\hat{\phi}=1-\bar{X}/(n\hat{p})\end{array}\right.$ (21)

where $\bar{X}^{+}$ is the average of strictly positive $X$’s. When
$\phi_{0}$, $p_{0}$ are estimated by $\hat{\phi}$, $\hat{p}$, the control
limits of the Shewhart chart are denoted as $\widehat{LCL}_{ZIB}$,
$\widehat{UCL}_{ZIB}$. Similarly, if the process parameters are estimated by
$\tilde{\phi}$, $\tilde{p}$, the respective limits are denoted as
$\widetilde{LCL}_{ZIB}$, $\widetilde{UCL}_{ZIB}$. Therefore, the control
limits of the ZIB-Shewhart chart in Case U are calculated by simply replacing
the (uknownn) values $\phi_{0}$ and $p_{0}$ with their respective estimates.
Then, practitioners can proceed to the Phase II process monitoring.

The performance of the ZIB-Shewhart chart in Case U is evaluated in terms of
the unconditional $ARL$ and $SDRL$. The steps for their calculation are
identical to those provided in Section 3.1 for the ZIP-Shewhart chart, after
some straightforward but necessary modifications. Due to space economy, the
details are left to the readers.

##  4 Numerical Results

In this section, we present the results of an extensive simulation study,
using 50000 simulation runs (i.e. $T=50000$), regarding the IC performance of
the ZIP- and ZIB-Shewhart charts in Case U.

###  4.1 The Unconditional Performance of the ZIP-Shewhart Chart

In Tables 1 and 2 we present the IC $ARL$ and $SDRL$ values for the ZIP-
Shewhart chart in Case U, for
$m\in\left\\{100,200,500,1000,2000,5000\right\\}$,
$\lambda_{0}\in\left\\{1,2,4,5,6,8\right\\}$ and
$\phi\in\left\\{0.9,0.8,0.7\right\\}$. The MLE method has been used to get the
results in Table 1 while the MoM has been used for the results in Table 2. The
first line in each cell gives the $ARL$ while the second line gives the
$SDRL$. In the column ‘Case K’ we provide the $ARL$ and $SDRL$ values in Case
K while constant $L$, given in the respective column, has been determined in
Case K (in two decimals accuracy) so as the IC $ARL$ is as close as possible
to the nominal value $ARL_{0}=370.4$, a value that appears very frequently in
several SPC textbooks. It is well known (see, for example, Rakitzis and
Castagliola (2016)) that due to the discrete nature of the ZIP distribution it
is almost impossible to achieve exactly the desired $ARL_{0}$ value. From the
results in column ’Case K’ in Table 1 (and Table 2) we deduce that there are
cases where the IC $ARL$ in Case K is above 370.4 while for the remaining ones
it is below.

It is not difficult to see that for the particular combinations
$(\phi_{0},\lambda_{0})$, the IC $ARL$ values can be very different between
the Case K and Case U. For example, from the results in Table 1, when
$(\phi_{0},\lambda_{0})=(0.8,4)$ the IC $ARL$ ($SDRL$) in Case K is
$ARL=234.04$ ($SDRL=233.54$) while when $m=200$ we have $ARL=566.39$
($SDRL=1116.81$). Note also that for a relatively small Phase I sample (i.e.
for $m=100$ or 200), the $ARL$ and $SDRL$ are very large, especially the
latter. Actually, for large $\phi_{0}$ values (such as 0.9 or 0.8) and for
Phase I samples of size $m=100$, the $SDRL$ values are extremely large. This
is an indication of increased uncertainty regarding the actual IC performance
of the chart. As $m$ increases, the IC $ARL$ values tend to decrease, but not
in a monotonic way. This is common for attributes control charts with
estimated parameters (see, for example, Castagliola and Wu (2012), Rakitzis
and Castagliola (2016), Mamzeridou and Rakitzis (2023) and references,
therein). Note also that as $m$ increases, it is not necessary that the IC
$ARL$ and $SDRL$ obtained in Case U will converge to the respective IC $ARL$
and $SDRL$ values in Case K. Even for very large Phase I samples (such as for
$m=2000$ or 5000), the $ARL$ and $SDRL$ can be significantly different than
the respective values in Case K. See, for example, the case
$(\phi_{0},\lambda_{0})=(0.8,1)$ or $(\phi_{0},\lambda_{0})=(0.7,2)$. This
fact is also attributed to the discrete nature of the ZIP distribution.

Table 1: IC $ARL$ and $SDRL$ values for the ZIP-Shewhart Chart in Case U when $L$ is from Case K - MLE method $\phi_{0}$ | $\lambda_{0}$ | $L$ | 100 | 200 | 500 | 1000 | 2000 | 5000 | Case K
---|---|---|---|---|---|---|---|---|---
0.9 | 1 | 6.66 | 735.21 | 426.62 | 330.69 | 323.88 | 325.72 | 326.11 | 526.64
|  |  | 3164.42 | 1039.20 | 502.21 | 434.35 | 431.99 | 432.28 | 526.14
| 2 | 6.41 | 1268.37 | 617.42 | 435.46 | 395.69 | 392.04 | 393.24 | 189.92
|  |  | 11282.42 | 1739.44 | 694.71 | 518.63 | 489.55 | 489.74 | 189.42
| 4 | 5.61 | 1173.64 | 568.04 | 390.56 | 348.66 | 335.52 | 335.86 | 468.09
|  |  | 14863.58 | 1527.86 | 614.13 | 447.65 | 392.86 | 386.74 | 467.59
| 5 | 5.72 | 2775.20 | 1043.62 | 643.09 | 555.63 | 522.05 | 511.23 | 314.19
|  |  | 32197.04 | 3564.15 | 1118.23 | 752.59 | 624.91 | 589.38 | 313.69
| 6 | 5.31 | 1685.94 | 714.62 | 457.20 | 396.67 | 374.75 | 366.50 | 497.71
|  |  | 14686.21 | 2545.71 | 764.17 | 522.35 | 439.39 | 410.96 | 497.21
| 8 | 5.15 | 3201.76 | 991.22 | 575.55 | 486.03 | 450.12 | 432.52 | 292.56
|  |  | 45373.05 | 4210.97 | 1025.86 | 665.61 | 538.16 | 480.22 | 292.06
0.8 | 1 | 6.33 | 1605.20 | 1033.08 | 811.91 | 797.17 | 801.98 | 799.38 | 263.32
|  |  | 8203.23 | 2641.74 | 1245.20 | 1115.59 | 1118.12 | 1116.19 | 262.82
| 2 | 5.49 | 1561.49 | 943.59 | 726.60 | 700.33 | 696.85 | 692.86 | 301.87
|  |  | 7511.36 | 2157.01 | 1075.06 | 911.12 | 897.56 | 894.40 | 301.37
| 4 | 4.47 | 850.31 | 566.39 | 449.27 | 424.31 | 420.75 | 417.34 | 234.04
|  |  | 3305.00 | 1116.81 | 620.58 | 518.11 | 499.33 | 496.14 | 233.54
| 5 | 4.47 | 1536.53 | 927.90 | 707.73 | 650.36 | 633.37 | 634.67 | 365.09
|  |  | 6766.62 | 2016.17 | 1033.60 | 808.04 | 746.39 | 744.54 | 364.59
| 6 | 4.09 | 875.03 | 568.34 | 446.35 | 411.82 | 403.97 | 400.85 | 248.86
|  |  | 3104.88 | 1129.01 | 622.37 | 498.47 | 463.78 | 458.92 | 248.36
| 8 | 3.89 | 1092.12 | 667.40 | 502.29 | 459.47 | 444.99 | 440.59 | 289.74
|  |  | 4205.83 | 1436.17 | 717.51 | 562.35 | 505.80 | 494.03 | 289.24
0.7 | 1 | 5.18 | 785.25 | 587.16 | 535.49 | 538.28 | 537.27 | 539.31 | 175.55
|  |  | 2352.71 | 1179.19 | 757.51 | 474.96 | 747.22 | 748.72 | 175.05
| 2 | 4.5 | 693.97 | 532.31 | 465.54 | 462.64 | 468.50 | 467.07 | 201.24
|  |  | 1799.96 | 941.45 | 615.07 | 597.17 | 601.32 | 600.20 | 200.74
| 4 | 3.66 | 412.11 | 328.80 | 291.87 | 287.96 | 290.63 | 294.41 | 409.89
|  |  | 896.25 | 516.96 | 362.38 | 339.81 | 341.00 | 344.01 | 409.39
| 5 | 3.65 | 658.47 | 506.09 | 434.03 | 420.35 | 416.23 | 411.57 | 243.39
|  |  | 1584.07 | 832.45 | 552.26 | 497.58 | 490.16 | 485.96 | 242.89
| 6 | 3.34 | 408.37 | 327.86 | 285.55 | 276.14 | 276.70 | 280.81 | 377.61
|  |  | 863.19 | 514.41 | 353.55 | 316.93 | 314.09 | 317.52 | 377.11
| 8 | 3.17 | 475.63 | 370.34 | 318.53 | 304.70 | 303.99 | 306.20 | 404.97
|  |  | 1114.49 | 594.23 | 398.22 | 347.88 | 338.69 | 340.27 | 404.47
Table 2: IC $ARL$ and $SDRL$ values for the ZIP-Shewhart Chart in Case U when $L$ is from Case K - MoM $\phi_{0}$ | $\lambda_{0}$ | $L$ | 100 | 200 | 500 | 1000 | 2000 | 5000 | Case K
---|---|---|---|---|---|---|---|---|---
0.9 | 1 | 6.66 | 883.14 | 427.10 | 328.68 | 318.96 | 322.32 | 326.28 | 526.64
|  |  | 12359.71 | 1145.97 | 518.36 | 432.19 | 429.41 | 432.41 | 526.14
| 2 | 6.41 | 1281.62 | 624.01 | 435.30 | 398.50 | 390.07 | 389.88 | 189.92
|  |  | 13991.43 | 1953.22 | 708.40 | 527.32 | 488.35 | 486.97 | 189.42
| 4 | 5.61 | 1462.50 | 577.83 | 391.60 | 348.63 | 335.95 | 335.42 | 468.09
|  |  | 58219.25 | 1993.99 | 628.37 | 452.55 | 394.96 | 386.37 | 467.59
| 5 | 5.72 | 2839.46 | 1068.15 | 653.69 | 558.32 | 521.99 | 511.90 | 314.19
|  |  | 31349.76 | 3975.49 | 1163.99 | 762.75 | 628.38 | 590.31 | 313.69
| 6 | 5.31 | 1759.65 | 721.64 | 458.78 | 398.31 | 373.00 | 367.80 | 497.71
|  |  | 15155.06 | 2718.01 | 775.81 | 533.86 | 439.46 | 412.17 | 497.21
| 8 | 5.15 | 3049.76 | 1022.58 | 578.72 | 488.46 | 448.43 | 433.13 | 292.56
|  |  | 36613.36 | 4224.83 | 1056.43 | 674.30 | 536.22 | 481.14 | 292.06
0.8 | 1 | 6.33 | 1920.78 | 1074.68 | 814.54 | 794.96 | 796.77 | 802.86 | 263.32
|  |  | 13635.01 | 3116.44 | 1285.25 | 1116.00 | 1114.25 | 1118.77 | 262.82
| 2 | 5.49 | 1647.73 | 984.21 | 727.80 | 696.39 | 694.18 | 696.20 | 301.87
|  |  | 9176.57 | 2471.15 | 1113.19 | 915.94 | 895.45 | 897.04 | 301.37
| 4 | 4.47 | 913.25 | 580.55 | 453.29 | 422.83 | 419.63 | 417.46 | 234.04
|  |  | 3924.83 | 1208.57 | 638.30 | 518.07 | 498.36 | 496.24 | 233.54
| 5 | 4.47 | 1670.69 | 948.22 | 713.63 | 652.88 | 637.28 | 636.91 | 365.09
|  |  | 9529.34 | 2154.13 | 1047.45 | 822.13 | 750.63 | 746.48 | 364.59
| 6 | 4.09 | 941.12 | 583.22 | 448.94 | 412.70 | 402.90 | 400.97 | 248.86
|  |  | 3628.02 | 1268.77 | 632.52 | 501.68 | 463.91 | 459.03 | 248.36
| 8 | 3.89 | 1157.29 | 684.89 | 509.14 | 464.15 | 445.14 | 441.07 | 289.74
|  |  | 4884.43 | 1528.66 | 730.69 | 571.11 | 506.42 | 494.43 | 289.24
0.7 | 1 | 5.18 | 848.42 | 601.17 | 532.47 | 536.05 | 533.64 | 539.36 | 175.55
|  |  | 3613.84 | 1268.09 | 753.96 | 746.32 | 744.54 | 748.75 | 175.05
| 2 | 4.5 | 744.48 | 541.96 | 468.19 | 464.60 | 466.30 | 465.76 | 201.24
|  |  | 2272.43 | 1000.84 | 631.23 | 598.28 | 599.60 | 599.18 | 200.74
| 4 | 3.66 | 428.65 | 337.05 | 291.65 | 287.69 | 289.70 | 293.15 | 409.89
|  |  | 1036.65 | 549.53 | 367.38 | 339.81 | 340.25 | 343.01 | 409.39
| 5 | 3.65 | 698.85 | 514.16 | 437.42 | 419.80 | 416.28 | 410.79 | 243.39
|  |  | 1788.67 | 863.89 | 563.35 | 497.57 | 490.20 | 485.24 | 242.89
| 6 | 3.34 | 432.39 | 333.83 | 286.85 | 278.06 | 278.10 | 280.96 | 377.61
|  |  | 1015.94 | 532.35 | 357.67 | 320.08 | 315.37 | 317.64 | 377.11
| 8 | 3.17 | 509.23 | 377.29 | 322.06 | 306.22 | 303.69 | 305.87 | 404.97
|  |  | 1291.78 | 613.58 | 407.06 | 351.84 | 338.65 | 339.99 | 404.47

The results so far show that if we want to apply the ZIP-Shewhart chart in the
practice, then we must collect a very large Phase I sample, even larger than
5000 preliminary observations, in order to have similar performance between
the Case K and Case U. Since this is not always possible in practice, we
provide in Tables 3 (MLE method) and 4 (MoM) the ‘adjusted’ $L^{*}$ values
(see column ‘$L^{*}$’ in the tables) which have been obtained so as the IC
$ARL$ in Case U is as close as possible to IC $ARL$ in Case K, for a given
size $m$ for the Phase I sample. Note also that any other value (instead of
the IC $ARL$ in Case K) can be used instead, according to practitioners needs.
Once the $L^{*}$ values has been obtained, we calculate the respective IC
$ARL$ and $SDRL$ values when the constant $L^{*}$ is used, instead of $L$.

The $L^{*}$ have been determined with a two decimals accuracy, using 50000
simulation runs. The steps of the algorithmic procedure are given at Algorithm
2.

Algorithm 2 Determination of the adjusted constant $L^{*}$ \- ZIP process

1:Choose the nominal IC $ARL$ value, say $ARL_{0}$.

2:Collect a Phase I sample of size $m$ from a $ZIP(\phi_{0},\lambda_{0})$
process.

3:Use the Phase I sample and estimate $\phi_{0}$, $\lambda_{0}$ by using the
MLE method (and get the $\hat{\phi}$, $\hat{\lambda}$ estimates).

4:Set the initial value $L\leftarrow 0.01$

5:Given the values $L$, $\hat{\phi}$ and $\hat{\lambda}$, determine
$\widehat{UCL}_{ZIP}$, $\widehat{LCL}_{ZIP}$ from equations (4) and (5) and
calculate the IC $ARL$.

6:Repeat Steps 2-5 $T$ times and calculate the unconditional IC $ARL$ as the
average of the $T$ values $1/\left(1-\hat{\beta}\right)$.

7:Compare the $ARL$ obtained in step 6 with the $ARL_{0}$. If $|ARL-
ARL_{0}|/ARL_{0}>0.05$ then increase $L$ by 0.01 and go back to Step 1.
Otherwise, select $L$ as the ‘adjusted’ $L^{*}$ value.

For example, in the case $(\phi_{0},\lambda_{0})=(0.8,4)$, the IC $ARL$
($SDRL$) in Case K is equal to $ARL=234.04$ ($SDRL=233.54$) with $L=4.47$.
Here $ARL_{0}=234.04$. When the size of the Phase I sample is $m=200$, then
the ‘adjusted’ value for the chart parameter is $L^{*}=4.02$ (when the process
parameters are estimated via the maximum likelihood estimation method), which
gives IC $ARL=234.34$ and $SDRL=390.11$. Recall that for $m=200$ and $L=4.47$
the $ARL=566.39$ and $SDRL=1116.81$. Therefore, the use of the ‘adjusted’
$L^{*}$ value results in an IC $ARL$ value is much closer to the nominal
$ARL_{0}$ value while there is also a significant reduction in the $SDRL$
value.

Table 3: ‘Adjusted’ $L^{*}$ Values for the ZIP-Shewhart Chart - MLE method |  | $\phi_{0}=0.9$ | $\phi_{0}=0.8$ | $\phi_{0}=0.7$
---|---|---|---|---
$\lambda_{0}$ | $m$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$
1 | 100 | 6.54 | 526.64 | 1914.88 | 4.86 | 263.22 | 650.62 | 4.10 | 179.75 | 374.91
| 200 | 6.90 | 528.91 | 1390.08 | 5.12 | 262.39 | 494.63 | 4.26 | 175.45 | 276.87
| 500 | 7.33 | 527.17 | 947.08 | 5.30 | 263.03 | 358.17 | 4.36 | 174.57 | 202.59
| 1000 | 7.51 | 529.11 | 739.50 | 5.38 | 263.06 | 288.63 | 4.41 | 175.39 | 177.76
| 2000 | 7.64 | 527.05 | 581.89 | 5.43 | 263.30 | 264.45 | 4.44 | 175.54 | 175.05
| 5000 | 7.67 | 526.33 | 526.29 | 5.64 | 263.34 | 262.94 | 4.44 | 175.55 | 175.05
2 | 100 | 4.98 | 189.51 | 526.49 | 4.53 | 300.65 | 806.72 | 3.80 | 199.34 | 400.56
| 200 | 5.30 | 189.66 | 360.71 | 4.73 | 304.39 | 543.58 | 3.92 | 202.41 | 309.87
| 500 | 5.52 | 190.31 | 265.89 | 4.86 | 300.18 | 406.96 | 4.00 | 201.09 | 243.50
| 1000 | 5.61 | 190.35 | 227.59 | 4.92 | 301.04 | 344.89 | 4.04 | 201.66 | 212.20
| 2000 | 5.67 | 189.86 | 200.66 | 4.96 | 302.01 | 309.55 | 4.05 | 201.21 | 201.62
| 5000 | 5.74 | 189.98 | 189.86 | 4.98 | 301.87 | 301.42 | 4.05 | 201.24 | 200.74
4 | 100 | 5.14 | 473.58 | 3861.22 | 3.88 | 235.33 | 579.53 | 3.66 | 409.93 | 924.35
| 200 | 5.49 | 463.95 | 1431.24 | 4.02 | 234.34 | 390.11 | 3.75 | 405.19 | 654.51
| 500 | 5.74 | 464.46 | 753.44 | 4.12 | 232.37 | 302.78 | 3.82 | 406.40 | 520.92
| 1000 | 5.84 | 466.90 | 618.94 | 4.15 | 234.95 | 273.28 | 3.85 | 408.91 | 460.79
| 2000 | 5.89 | 465.12 | 547.49 | 4.17 | 233.91 | 246.11 | 3.86 | 407.91 | 418.67
| 5000 | 5.94 | 468.77 | 487.91 | 4.19 | 234.04 | 234.12 | 3.88 | 409.90 | 409.54
5 | 100 | 4.73 | 323.72 | 1673.04 | 3.92 | 365.10 | 993.44 | 3.28 | 242.37 | 487.02
| 200 | 5.04 | 314.12 | 726.86 | 4.06 | 364.24 | 663.87 | 3.36 | 242.89 | 364.17
| 500 | 5.24 | 313.38 | 480.98 | 4.16 | 366.44 | 499.38 | 3.41 | 241.53 | 297.82
| 1000 | 5.32 | 313.61 | 400.84 | 4.20 | 367.29 | 442.19 | 3.44 | 245.08 | 272.32
| 2000 | 5.36 | 311.84 | 359.94 | 4.22 | 365.72 | 398.56 | 3.45 | 243.57 | 249.88
| 5000 | 5.40 | 314.64 | 327.18 | 4.24 | 365.54 | 368.39 | 3.46 | 243.46 | 243.16
6 | 100 | 4.81 | 499.73 | 2733.80 | 3.62 | 250.20 | 689.91 | 3.31 | 381.12 | 826.60
| 200 | 4.79 | 492.48 | 3018.95 | 3.75 | 250.10 | 443.31 | 3.39 | 372.96 | 594.64
| 500 | 5.12 | 495.15 | 1425.54 | 3.83 | 249.11 | 326.73 | 3.45 | 380.46 | 482.38
| 1000 | 5.45 | 493.99 | 663.90 | 3.86 | 248.56 | 292.40 | 3.47 | 378.22 | 431.74
| 2000 | 5.50 | 499.64 | 597.13 | 3.88 | 248.15 | 267.48 | 3.48 | 376.75 | 394.47
| 5000 | 5.53 | 496.66 | 530.07 | 3.89 | 248.49 | 250.36 | 3.49 | 377.58 | 377.91
8 | 100 | 4.33 | 299.65 | 1500.19 | 3.47 | 289.30 | 796.77 | 3.12 | 404.77 | 886.12
| 200 | 4.60 | 293.34 | 777.60 | 3.60 | 293.82 | 548.37 | 3.20 | 409.21 | 670.61
| 500 | 4.80 | 292.26 | 463.56 | 3.68 | 289.34 | 390.57 | 3.25 | 407.76 | 519.92
| 1000 | 4.87 | 292.58 | 378.29 | 3.71 | 289.76 | 344.97 | 3.27 | 409.30 | 473.06
| 2000 | 4.91 | 293.21 | 340.75 | 3.73 | 292.57 | 323.01 | 3.28 | 406.63 | 433.95
| 5000 | 4.94 | 294.13 | 313.23 | 3.74 | 289.99 | 295.78 | 3.29 | 405.99 | 408.78
Table 4: ‘Adjusted’ $L^{*}$ Values for the ZIP-Shewhart Chart - MoM |  | $\phi_{0}=0.9$ | $\phi_{0}=0.8$ | $\phi_{0}=0.7$
---|---|---|---|---
$\lambda_{0}$ | $m$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$
1 | 100 | 6.25 | 531.07 | 3095.24 | 4.86 | 267.77 | 857.32 | 4.09 | 175.79 | 391.94
| 200 | 6.90 | 543.54 | 1649.93 | 5.11 | 263.45 | 508.96 | 4.24 | 176.52 | 287.04
| 500 | 7.33 | 529.79 | 951.57 | 5.30 | 264.72 | 369.75 | 4.36 | 175.91 | 209.19
| 1000 | 7.51 | 532.30 | 756.62 | 5.38 | 264.53 | 296.31 | 4.42 | 175.79 | 180.14
| 2000 | 7.63 | 527.60 | 592.97 | 5.44 | 263.61 | 265.94 | 4.44 | 175.55 | 175.19
| 5000 | 7.68 | 526.70 | 528.12 | 5.32 | 263.32 | 262.82 | 4.25 | 175.55 | 175.05
2 | 100 | 4.98 | 193.14 | 792.41 | 4.52 | 306.59 | 855.64 | 3.80 | 201.34 | 419.99
| 200 | 5.30 | 190.68 | 384.17 | 4.72 | 306.55 | 578.28 | 3.92 | 204.84 | 325.62
| 500 | 5.53 | 192.77 | 273.03 | 4.86 | 303.33 | 422.58 | 4.00 | 202.36 | 250.62
| 1000 | 5.61 | 190.45 | 231.20 | 4.92 | 304.04 | 357.06 | 4.04 | 201.89 | 215.39
| 2000 | 5.67 | 190.04 | 202.49 | 4.96 | 302.50 | 312.96 | 4.06 | 201.38 | 202.14
| 5000 | 5.71 | 189.98 | 189.95 | 4.97 | 301.87 | 301.48 | 4.04 | 201.24 | 400.74
4 | 100 | 5.12 | 468.23 | 2383.37 | 3.86 | 235.30 | 634.38 | 3.65 | 424.18 | 1115.79
| 200 | 5.48 | 470.19 | 3398.92 | 4.02 | 237.56 | 416.71 | 3.75 | 412.83 | 687.30
| 500 | 5.75 | 471.20 | 782.49 | 4.11 | 234.15 | 309.95 | 3.82 | 412.84 | 539.78
| 1000 | 5.84 | 468.29 | 631.06 | 4.15 | 234.46 | 274.23 | 3.85 | 411.61 | 470.61
| 2000 | 5.90 | 471.43 | 561.97 | 4.18 | 235.88 | 250.54 | 3.87 | 411.04 | 425.87
| 5000 | 5.94 | 468.75 | 489.51 | 4.19 | 234.09 | 234.39 | 3.88 | 409.90 | 409.62
5 | 100 | 4.71 | 320.47 | 1739.25 | 3.90 | 371.43 | 1109.44 | 3.26 | 237.90 | 480.98
| 200 | 5.02 | 314.95 | 770.66 | 4.06 | 373.64 | 710.68 | 3.36 | 248.60 | 381.95
| 500 | 5.25 | 320.30 | 500.50 | 4.16 | 366.83 | 504.38 | 3.42 | 248.55 | 309.66
| 1000 | 5.33 | 318.47 | 411.56 | 4.20 | 369.94 | 450.50 | 3.44 | 247.02 | 278.25
| 2000 | 5.37 | 316.00 | 367.03 | 4.22 | 366.28 | 401.06 | 3.45 | 244.04 | 251.96
| 5000 | 5.40 | 314.45 | 327.80 | 4.24 | 365.33 | 368.72 | 3.46 | 243.46 | 243.30
6 | 100 | 4.79 | 510.34 | 4858.52 | 3.61 | 253.05 | 691.00 | 3.30 | 386.09 | 883.67
| 200 | 5.13 | 519.81 | 1693.26 | 3.74 | 249.54 | 441.93 | 3.39 | 382.38 | 621.35
| 500 | 5.37 | 509.23 | 886.18 | 3.83 | 251.35 | 333.00 | 3.45 | 384.21 | 492.48
| 1000 | 5.45 | 500.16 | 679.99 | 3.86 | 249.25 | 294.17 | 3.47 | 380.40 | 438.07
| 2000 | 5.50 | 500.60 | 601.97 | 3.88 | 250.16 | 272.42 | 3.49 | 383.03 | 405.29
| 5000 | 5.54 | 502.74 | 539.86 | 3.90 | 249.53 | 252.24 | 3.50 | 378.26 | 379.39
8 | 100 | 4.29 | 299.06 | 5744.17 | 3.46 | 293.92 | 923.36 | 3.10 | 397.00 | 952.52
| 200 | 4.60 | 296.97 | 784.66 | 3.60 | 297.99 | 576.82 | 3.18 | 392.39 | 645.46
| 500 | 4.80 | 294.37 | 468.89 | 3.68 | 293.14 | 400.61 | 3.24 | 400.23 | 513.15
| 1000 | 4.87 | 293.66 | 381.99 | 3.71 | 291.57 | 349.50 | 3.26 | 400.55 | 463.44
| 2000 | 4.91 | 295.21 | 344.15 | 3.73 | 292.42 | 323.03 | 3.28 | 407.42 | 436.28
| 5000 | 4.94 | 295.41 | 315.50 | 3.74 | 290.44 | 296.90 | 3.29 | 405.94 | 408.91

Before closing this section it is worth mentioning that Tables 1-4 offer a
head-to-head comparison between the MLEs and the MEs. Recall that for the MEs
there are closed-form expressions while MLEs require the use of a non-linear
equation solver and can only be obtained numerically. The results show that
for Phase I samples of size $m\geq 500$, the two methods provide (almost)
identical results. When $m=100$ or 200, the respective $ARL$ and $SDRL$ values
can be very different. According to Beckett et al. (2014); Wagh and Kamalja
(2018), the MLEs are superior than the estimates obtained by the MoM, in terms
of mean square error and bias, especially for small sample sizes. Therefore,
we would suggest the use of MLEs when very large Phase I samples are not
available. However, for larger Phase I samples (e.g. when $m>1000$) the MoM
estimators of $\phi_{0}$, $\lambda_{0}$ might be more appealing to
practitioners due to their simplicity in calculation.

###  4.2 The Unconditional Performance of the ZIB-Shewhart Chart

Tables 5-6 and Tables 7-8 are counterparts to Tables 1-2 and Tables 3-4 but
for the ZIB-Shewhart chart. In Tables 5 and 6 we present the IC $ARL$ and
$SDRL$ values for the ZIB-Shewhart chart, for $n\in\left\\{100,250\right\\}$,
$p_{0}\in\left\\{0.01,0.02,0.030\right\\}$,
$m\in\left\\{100,200,500,1000,2000,5000\right\\}$ and
$\phi\in\\{0.9,0.8,0.7\\}$. The value of $L$ has been determined in order to
achieve an IC $ARL$ value in Case K that it is as close as possible to the
nominal $ARL_{0}=370.4$. Note also that the results in Table 5 have been
obtained via the MLE method while for those in Table 6 we used the MoM. Again,
the values of the achieved $ARL_{0}$ can be very different than the nominal
370.4 value. As for the case of ZIP-Shewhart chart, this is attributed to the
discrete nature of the ZIB distribution and the large value of the zero-
inflation parameter $\phi_{0}$.

For example, for $\phi_{0}=0.8$, $p_{0}=0.01$, $n=100$ and $L=6.35$, the
control limits are obtained from equations (10) and (11) and they equal
$UCL_{ZIB}=3$, $LCL_{ZIB}=0$. Therefore, using equation (12), the FAR equals
$1-\beta=1-F_{ZIB}(3\left|0.8,100,0.01\right.)+F_{ZIB}(0-1\left|0.8,100,0.01\right.)\approx
0.0036748+0=0.0036748$ and the $ARL_{0}=1/0.0036748\approx 272.12$.

Conclusions from Tables 5 and 6 are similar to those from Tables 1 and 2.
Specifically, the IC $ARL$ values can be very different between Case K and
Case U. For example, if $(\phi_{0},n,p_{0})=(0.9,250,0.01)$ then the IC $ARL$
and $SDRL$ are 242.82 and 242.32, respectively. But, in Case U and for
$m=200$, the IC $ARL$ and $SDRL$ are 828.37 and 2962.55, respectively.
Clearly, these values are much larger than the respective ones in Case K,
especially the IC $SDRL$. Therefore, very large Phase I samples are needed
(e.g. $m\geq 1000$) while there are cases where even $m=5000$ preliminary
observations might not be enough in order to have similar performance between
Case K and Case U. See, for example, the cases
$(\phi_{0},n,p_{0})=(0.8,100,0.01)$ or $(\phi_{0},n,p_{0})=(0.9,100,0.02)$.
However, as the size $m$ increases, the IC $SDRL$ reduces while the IC $ARL$
converges (not monotonically) to a specific value, which is not necessarily
the same with the IC $ARL$ under Case K.

Table 5: IC $ARL$ and $SDRL$ values for the ZIB-Shewhart Chart in Case U when $L$ is from Case K - MLE method $\phi_{0}$ | $p_{0}$ | $n$ | $L$ | 100 | 200 | 500 | 1000 | 2000 | 5000 | Case K
---|---|---|---|---|---|---|---|---|---|---
0.9 | 0.01 | 100 | 6.68 | 905.96 | 450.09 | 340.95 | 332.53 | 334.73 | 337.79 | 544.25
|  |  |  | 5067.77 | 1205.60 | 518.39 | 446.73 | 446.30 | 448.57 | 543.75
|  | 250 | 6.38 | 1804.08 | 828.37 | 551.49 | 490.08 | 478.06 | 475.08 | 242.82
|  |  |  | 16092.61 | 2962.55 | 916.65 | 654.97 | 590.73 | 586.15 | 242.32
0.8 | 0.01 | 100 | 6.35 | 1959.66 | 1106.40 | 870.21 | 854.52 | 851.31 | 858.41 | 272.12
|  |  |  | 11762.09 | 2957.93 | 1349.42 | 1197.39 | 1193.81 | 1199.00 | 271.62
|  | 250 | 5.31 | 1845.05 | 1117.18 | 836.70 | 793.15 | 788.18 | 783.97 | 364.92
|  |  |  | 9380.28 | 2617.58 | 1248.05 | 1028.49 | 1002.35 | 998.90 | 364.42
0.7 | 0.01 | 100 | 5.19 | 844.34 | 617.27 | 562.32 | 565.09 | 561.93 | 558.17 | 181.42
|  |  |  | 2883.62 | 1213.13 | 799.13 | 793.95 | 791.61 | 788.79 | 180.91
|  | 250 | 4.35 | 815.67 | 604.66 | 528.87 | 520.38 | 513.14 | 508.91 | 243.28
|  |  |  | 2210.39 | 1068.48 | 700.18 | 664.77 | 657.86 | 654.25 | 242.78
0.9 | 0.02 | 100 | 6.43 | 1390.59 | 658.03 | 463.27 | 419.49 | 413.94 | 413.81 | 196.73
|  |  |  | 9792.72 | 1864.65 | 748.52 | 552.77 | 522.49 | 521.12 | 196.23
|  | 250 | 5.73 | 3301.38 | 1126.98 | 695.83 | 596.43 | 555.99 | 542.99 | 329.22
|  |  |  | 76501.60 | 3833.54 | 1231.11 | 825.55 | 668.75 | 630.13 | 328.72
0.8 | 0.02 | 100 | 5.51 | 1752.68 | 1059.84 | 800.56 | 773.11 | 768.20 | 766.16 | 322.92
|  |  |  | 8044.89 | 2697.97 | 1211.15 | 1018.73 | 1000.71 | 999.11 | 322.42
|  | 250 | 4.04 | 546.84 | 367.64 | 300.80 | 284.05 | 281.48 | 284.06 | 390.77
|  |  |  | 2701.38 | 675.68 | 402.04 | 338.08 | 323.91 | 325.42 | 390.27
0.7 | 0.02 | 100 | 4.51 | 771.87 | 581.34 | 509.49 | 501.80 | 499.72 | 489.92 | 215.28
|  |  |  | 2145.68 | 1034.41 | 686.15 | 659.35 | 657.09 | 648.97 | 214.78
|  | 250 | 3.66 | 730.52 | 556.84 | 475.17 | 458.28 | 456.62 | 451.26 | 260.51
|  |  |  | 1834.69 | 948.22 | 613.03 | 544.71 | 540.56 | 535.75 | 260.01
0.9 | 0.03 | 100 | 6.38 | 3195.35 | 1230.92 | 756.34 | 657.13 | 624.07 | 620.12 | 320.23
|  |  |  | 55772.52 | 4905.18 | 1345.95 | 909.57 | 774.01 | 759.32 | 319.73
|  | 250 | 5.09 | 2297.93 | 808.28 | 484.48 | 414.70 | 388.05 | 375.15 | 248.86
|  |  |  | 34034.94 | 2666.92 | 834.65 | 557.19 | 460.84 | 417.27 | 248.36
0.8 | 0.03 | 100 | 4.51 | 570.29 | 398.58 | 330.08 | 319.25 | 322.23 | 326.26 | 470.64
|  |  |  | 1891.84 | 746.76 | 445.74 | 390.33 | 389.40 | 392.55 | 470.14
|  | 250 | 3.86 | 935.83 | 576.64 | 439.76 | 405.54 | 391.48 | 387.97 | 252.12
|  |  |  | 3658.73 | 1244.32 | 619.49 | 494.25 | 446.75 | 439.04 | 251.61
0.7 | 0.03 | 100 | 4.27 | 1083.34 | 802.81 | 680.58 | 667.35 | 668.83 | 668.83 | 313.76
|  |  |  | 3109.93 | 1467.40 | 910.28 | 842.23 | 841.27 | 840.69 | 313.26
|  | 250 | 3.4 | 1035.11 | 760.02 | 621.56 | 589.34 | 578.33 | 567.52 | 363.24
|  |  |  | 2753.87 | 1373.60 | 816.73 | 697.60 | 667.16 | 656.36 | 362.74
Table 6: IC $ARL$ and $SDRL$ values for the ZIB-Shewhart Chart in Case U when $L$ is from Case K - MoM $\phi_{0}$ | $p_{0}$ | $n$ | $L$ | 100 | 200 | 500 | 1000 | 2000 | 5000 | Case K
---|---|---|---|---|---|---|---|---|---|---
0.9 | 0.01 | 100 | 6.68 | 819.19 | 442.89 | 337.97 | 329.77 | 330.31 | 335.92 | 544.25
|  |  |  | 5757.19 | 1528.21 | 537.91 | 449.36 | 442.94 | 447.19 | 543.75
|  | 250 | 6.38 | 1964.63 | 813.32 | 553.84 | 490.51 | 476.76 | 475.55 | 242.82
|  |  |  | 38479.06 | 2549.29 | 952.35 | 663.62 | 590.62 | 586.55 | 242.32
0.8 | 0.01 | 100 | 6.35 | 2066.18 | 1128.33 | 856.90 | 844.78 | 849.41 | 853.56 | 272.12
|  |  |  | 14747.90 | 3252.57 | 1376.88 | 1196.22 | 1192.42 | 1195.46 | 271.62
|  | 250 | 6 | 1907.18 | 1126.10 | 839.75 | 789.24 | 783.95 | 779.37 | 364.92
|  |  |  | 11480.70 | 2864.85 | 1286.38 | 1029.17 | 998.91 | 995.10 | 364.42
0.7 | 0.01 | 100 | 5.19 | 859.75 | 620.89 | 557.01 | 555.93 | 557.25 | 553.53 | 181.42
|  |  |  | 3177.49 | 1270.84 | 798.42 | 787.10 | 788.10 | 785.27 | 180.91
|  | 250 | 4.35 | 814.89 | 609.44 | 525.10 | 517.14 | 512.98 | 503.76 | 243.28
|  |  |  | 2371.18 | 1135.87 | 710.05 | 662.48 | 657.73 | 649.79 | 242.78
0.9 | 0.02 | 100 | 6.43 | 1411.89 | 664.56 | 460.10 | 417.44 | 411.19 | 410.55 | 196.73
|  |  |  | 19130.12 | 2215.13 | 772.55 | 558.28 | 520.09 | 518.42 | 196.23
|  | 250 | 5.73 | 3139.49 | 1149.55 | 691.13 | 594.82 | 554.54 | 542.81 | 329.22
|  |  |  | 36337.64 | 4337.88 | 1249.93 | 826.06 | 673.75 | 629.65 | 328.72
0.8 | 0.02 | 100 | 5.51 | 1805.57 | 1050.28 | 804.90 | 763.27 | 764.15 | 770.48 | 322.92
|  |  |  | 8924.58 | 2581.10 | 1249.33 | 1010.73 | 997.52 | 1002.50 | 322.42
|  | 250 | 4.04 | 524.27 | 364.56 | 301.95 | 283.03 | 281.73 | 284.55 | 390.77
|  |  |  | 1687.15 | 693.27 | 412.39 | 337.16 | 324.26 | 325.83 | 390.27
0.7 | 0.02 | 100 | 4.51 | 781.04 | 586.60 | 505.38 | 500.09 | 498.89 | 490.52 | 215.28
|  |  |  | 2441.09 | 1146.70 | 684.72 | 657.40 | 656.41 | 649.48 | 214.78
|  | 250 | 3.66 | 728.28 | 556.02 | 473.93 | 457.86 | 455.61 | 451.69 | 260.51
|  |  |  | 1916.02 | 959.71 | 618.42 | 546.08 | 539.76 | 536.13 | 260.01
0.9 | 0.03 | 100 | 6.38 | 3265.02 | 1217.89 | 749.80 | 653.31 | 624.05 | 618.16 | 320.23
|  |  |  | 47087.82 | 4698.07 | 1359.02 | 923.07 | 776.75 | 757.65 | 319.73
|  | 250 | 5.09 | 2174.34 | 796.58 | 483.49 | 416.57 | 386.64 | 373.77 | 248.86
|  |  |  | 22699.92 | 3104.88 | 854.60 | 570.90 | 460.04 | 416.15 | 248.36
0.8 | 0.03 | 100 | 4.51 | 571.83 | 398.43 | 329.15 | 318.58 | 319.97 | 324.40 | 470.64
|  |  |  | 2249.32 | 805.98 | 454.06 | 392.87 | 387.60 | 391.10 | 470.14
|  | 250 | 3.86 | 904.74 | 574.09 | 441.43 | 404.73 | 391.78 | 387.30 | 252.12
|  |  |  | 3503.86 | 1234.12 | 632.01 | 495.61 | 446.67 | 483.42 | 251.61
0.7 | 0.03 | 100 | 4.27 | 1111.46 | 805.86 | 680.79 | 668.27 | 666.85 | 668.61 | 313.76
|  |  |  | 3585.85 | 1501.58 | 922.51 | 845.54 | 839.90 | 841.10 | 313.26
|  | 250 | 3.4 | 1029.06 | 754.55 | 622.78 | 586.09 | 578.36 | 567.83 | 363.24
|  |  |  | 2848.15 | 1366.98 | 825.29 | 695.60 | 667.58 | 656.65 | 362.74

In Tables 7 and 8 we provide the ‘adjusted’ values $L^{*}$ of the chart’s
design parameter $L$, so as for a pre-specified size $m$ of the Phase I
sample, the IC $ARL$ under Case K is the closest possible to the IC $ARL$
under Case K. The $L^{*}$ has been determined with a two decimals accuracy,
using Monte Carlo simulation. The steps of the related algorithmic procedure
are similar to those presented in Section 3.1, for the case of the ZIP-
Shewhart chart. Some direct but necessary modifications are needed. The
results in both Tables 7 and 8 show that the use of the ‘adjusted’ $L^{*}$
results in IC $ARL$ values much closer to the respective value in Case K while
the IC $SDRL$ is much lower, compared to the case where the unadjusted $L$
value is used.

Table 7: ‘Adjusted’ $L^{*}$ Values for the ZIB-Shewhart Chart - MLE method |  |  | $\phi_{0}=0.9$ | $\phi_{0}=0.8$ | $\phi_{0}=0.7$
---|---|---|---|---|---
$p_{0}$ | $n$ | $m$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$
0.01 | 100 | 100 | 6.22 | 547.62 | 2898.06 | 4.83 | 275.82 | 734.92 | 4.10 | 187.96 | 416.23
|  | 200 | 6.86 | 553.26 | 1464.83 | 5.13 | 277.34 | 536.52 | 4.27 | 182.05 | 293.11
|  | 500 | 7.36 | 554.95 | 984.83 | 5.30 | 272.52 | 379.78 | 4.37 | 181.63 | 215.06
|  | 1000 | 7.53 | 550.44 | 783.88 | 5.40 | 272.71 | 300.60 | 4.43 | 181.48 | 184.54
|  | 2000 | 7.66 | 544.80 | 603.72 | 5.46 | 272.36 | 274.22 | 4.44 | 181.43 | 181.08
|  | 5000 | 7.73 | 544.28 | 544.97 | 5.34 | 272.12 | 271.62 | 4.21 | 181.42 | 180.91
0.02 | 100 | 100 | 4.99 | 196.97 | 718.36 | 4.55 | 326.95 | 914.88 | 3.83 | 218.21 | 455.28
|  | 200 | 5.30 | 197.17 | 380.91 | 4.74 | 324.82 | 606.08 | 3.94 | 216.92 | 338.19
|  | 500 | 5.54 | 197.61 | 277.31 | 4.88 | 324.36 | 450.06 | 4.03 | 217.99 | 267.40
|  | 1000 | 5.63 | 196.87 | 237.15 | 4.94 | 322.97 | 373.78 | 4.06 | 216.04 | 228.13
|  | 2000 | 5.70 | 197.22 | 208.97 | 4.98 | 323.02 | 331.20 | 4.08 | 215.36 | 215.65
|  | 5000 | 5.74 | 196.78 | 196.75 | 5.00 | 322.93 | 322.55 | 4.01 | 215.28 | 214.78
0.03 | 100 | 100 | 5.09 | 323.34 | 1293.61 | 4.42 | 477.90 | 1513.62 | 3.73 | 317.90 | 673.94
|  | 200 | 5.44 | 323.53 | 747.21 | 4.60 | 473.39 | 965.78 | 3.83 | 315.70 | 503.62
|  | 500 | 5.68 | 320.37 | 483.84 | 4.73 | 472.69 | 677.53 | 3.90 | 314.05 | 400.09
|  | 1000 | 5.77 | 320.28 | 415.16 | 4.78 | 471.26 | 576.32 | 3.94 | 316.69 | 349.47
|  | 2000 | 5.83 | 320.38 | 362.67 | 4.82 | 472.82 | 504.36 | 3.95 | 313.84 | 317.76
|  | 5000 | 5.88 | 320.43 | 324.91 | 4.85 | 470.93 | 471.65 | 3.96 | 313.76 | 313.26
0.01 | 250 | 100 | 5.05 | 248.87 | 943.69 | 4.47 | 375.30 | 1005.00 | 3.76 | 246.02 | 511.11
|  | 200 | 5.38 | 245.81 | 525.73 | 4.65 | 368.71 | 689.54 | 3.86 | 244.62 | 379.11
|  | 500 | 5.62 | 245.20 | 356.34 | 4.79 | 370.87 | 515.43 | 3.94 | 244.92 | 304.14
|  | 1000 | 5.70 | 243.80 | 304.10 | 4.83 | 365.06 | 432.10 | 3.97 | 243.45 | 260.65
|  | 2000 | 5.76 | 243.18 | 265.94 | 4.87 | 365.57 | 381.55 | 3.99 | 243.39 | 244.93
|  | 5000 | 5.81 | 243.10 | 244.36 | 4.90 | 364.99 | 364.83 | 3.99 | 243.28 | 242.78
0.02 | 250 | 100 | 4.74 | 337.48 | 1522.58 | 3.92 | 396.38 | 1240.43 | 3.29 | 262.28 | 539.24
|  | 200 | 5.04 | 332.37 | 875.33 | 4.07 | 393.34 | 748.01 | 3.37 | 262.76 | 397.89
|  | 500 | 5.25 | 331.23 | 511.43 | 4.17 | 393.15 | 537.20 | 3.43 | 266.14 | 331.43
|  | 1000 | 5.33 | 330.02 | 426.54 | 4.21 | 395.09 | 480.39 | 3.45 | 263.90 | 295.47
|  | 2000 | 5.38 | 331.36 | 385.70 | 4.23 | 392.28 | 427.76 | 3.46 | 261.44 | 268.93
|  | 5000 | 5.41 | 329.94 | 343.60 | 4.25 | 390.37 | 394.67 | 3.48 | 260.72 | 260.59
0.03 | 250 | 100 | 4.28 | 252.68 | 1350.49 | 3.44 | 252.94 | 674.34 | 3.10 | 359.07 | 821.73
|  | 200 | 4.54 | 250.29 | 646.02 | 3.57 | 260.90 | 480.26 | 3.17 | 351.75 | 562.90
|  | 500 | 4.73 | 250.57 | 390.05 | 3.65 | 256.79 | 344.24 | 3.23 | 362.81 | 461.89
|  | 1000 | 4.80 | 251.69 | 321.47 | 3.68 | 257.94 | 307.20 | 3.24 | 354.55 | 406.47
|  | 2000 | 4.84 | 251.71 | 290.24 | 3.69 | 252.70 | 276.63 | 3.26 | 362.99 | 384.36
|  | 5000 | 4.87 | 252.18 | 267.24 | 3.71 | 253.99 | 258.71 | 3.26 | 362.34 | 363.33
Table 8: ‘Adjusted’ $L^{*}$ Values for the ZIB-Shewhart Chart - MoM |  |  | $\phi_{0}=0.9$ | $\phi_{0}=0.8$ | $\phi_{0}=0.7$
---|---|---|---|---|---
$p_{0}$ | $n$ | $m$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$ | $L^{*}$ | $ARL$ | $SDRL$
0.01 | 100 | 100 | 6.22 | 549.55 | 8769.46 | 4.88 | 272.38 | 892.13 | 4.12 | 182.62 | 413.66
|  | 200 | 6.92 | 544.46 | 1711.07 | 5.12 | 272.35 | 542.85 | 4.26 | 182.66 | 307.14
|  | 500 | 7.35 | 544.48 | 970.99 | 5.31 | 273.55 | 386.41 | 4.38 | 182.00 | 218.65
|  | 1000 | 7.53 | 549.34 | 794.59 | 5.40 | 273.23 | 305.94 | 4.43 | 181.43 | 185.40
|  | 2000 | 7.66 | 545.77 | 613.90 | 5.46 | 272.41 | 274.95 | 4.45 | 181.46 | 181.25
|  | 5000 | 7.73 | 544.25 | 545.20 | 5.37 | 272.12 | 271.62 | 4.26 | 181.42 | 180.91
0.02 | 100 | 100 | 5.01 | 198.13 | 728.96 | 4.54 | 325.00 | 979.55 | 3.83 | 215.85 | 476.67
|  | 200 | 5.33 | 196.99 | 395.82 | 4.75 | 325.78 | 619.55 | 3.94 | 216.50 | 345.00
|  | 500 | 5.55 | 199.01 | 284.76 | 4.88 | 325.51 | 461.30 | 4.02 | 215.86 | 268.80
|  | 1000 | 5.63 | 197.01 | 240.24 | 4.94 | 324.76 | 382.56 | 4.06 | 216.52 | 231.72
|  | 2000 | 5.70 | 197.76 | 211.85 | 4.97 | 323.00 | 333.42 | 4.08 | 215.41 | 216.14
|  | 5000 | 5.73 | 196.74 | 196.76 | 5.00 | 322.95 | 322.62 | 4.01 | 215.28 | 214.78
0.03 | 100 | 100 | 5.10 | 323.72 | 1962.17 | 4.42 | 477.24 | 1577.82 | 3.73 | 320.25 | 747.63
|  | 200 | 5.45 | 323.21 | 829.87 | 4.60 | 472.48 | 1011.32 | 3.83 | 315.39 | 514.54
|  | 500 | 5.69 | 320.64 | 497.95 | 4.73 | 474.02 | 692.41 | 3.90 | 314.86 | 408.38
|  | 1000 | 5.77 | 320.75 | 420.67 | 4.78 | 471.66 | 584.45 | 3.94 | 317.75 | 356.21
|  | 2000 | 5.83 | 320.69 | 366.18 | 4.82 | 473.45 | 510.22 | 3.96 | 314.94 | 321.24
|  | 5000 | 5.87 | 320.24 | 325.46 | 4.83 | 470.68 | 471.69 | 3.96 | 313.78 | 313.31
0.01 | 250 | 100 | 5.07 | 244.13 | 891.46 | 4.46 | 364.97 | 1082.59 | 3.76 | 244.34 | 555.38
|  | 200 | 5.40 | 248.42 | 543.63 | 4.66 | 368.34 | 699.62 | 3.87 | 248.16 | 398.63
|  | 500 | 5.62 | 244.77 | 360.81 | 4.78 | 366.04 | 516.36 | 3.94 | 244.81 | 308.94
|  | 1000 | 5.70 | 243.84 | 307.25 | 4.84 | 367.57 | 442.50 | 3.97 | 243.55 | 264.50
|  | 2000 | 5.76 | 243.47 | 268.96 | 4.87 | 366.00 | 385.89 | 3.99 | 243.30 | 245.08
|  | 5000 | 5.81 | 243.20 | 244.94 | 4.90 | 365.03 | 365.05 | 3.96 | 243.28 | 242.78
0.02 | 250 | 100 | 4.75 | 344.06 | 1986.21 | 3.93 | 402.80 | 1373.22 | 3.29 | 262.04 | 565.00
|  | 200 | 5.04 | 332.03 | 840.28 | 4.07 | 392.60 | 756.94 | 3.37 | 262.55 | 410.71
|  | 500 | 5.26 | 332.81 | 522.39 | 4.17 | 393.90 | 550.32 | 3.43 | 265.68 | 333.49
|  | 1000 | 5.34 | 332.74 | 432.10 | 4.21 | 395.62 | 482.49 | 3.45 | 264.35 | 298.61
|  | 2000 | 5.38 | 330.73 | 386.37 | 4.23 | 392.27 | 430.20 | 3.46 | 261.68 | 270.50
|  | 5000 | 5.41 | 330.15 | 345.08 | 4.25 | 391.27 | 394.91 | 3.47 | 260.55 | 260.38
0.03 | 250 | 100 | 4.30 | 259.09 | 1453.56 | 3.45 | 258.11 | 842.59 | 3.10 | 358.13 | 868.61
|  | 200 | 4.55 | 250.38 | 628.92 | 3.57 | 258.28 | 484.50 | 3.18 | 362.44 | 592.21
|  | 500 | 4.73 | 249.27 | 389.73 | 3.65 | 256.53 | 345.74 | 3.22 | 351.99 | 449.17
|  | 1000 | 4.80 | 251.08 | 321.89 | 3.68 | 257.24 | 307.31 | 3.24 | 354.47 | 407.83
|  | 2000 | 4.84 | 251.44 | 290.47 | 3.69 | 253.20 | 278.18 | 3.25 | 356.52 | 377.35
|  | 5000 | 4.87 | 252.08 | 267.51 | 3.71 | 254.03 | 259.04 | 3.26 | 362.22 | 363.50

##  5 Out-of-Control Performance of the Charts in Case U

In this section, we investigate the OOC performance of the ZIP- and ZIB-
Shewhart charts. It is expected that when the ‘adjusted’ constant $L^{*}$ is
used, there will be differences in the detection ability of the chart. These
differences also depend on the size $m$ of the Phase I sample as well as on
the magnitude of shift(s) in process parameters. Again, we use Monte Carlo
simulation and the $ARL$, $SDRL$ values are provided for several shifts in
process parameters. Due to space economy, only the results in the case of MLE
method are presented.

###  5.1 Out-of-Control Performance of the ZIP-Charts in Case U

In Table 9, we give the $ARL$ and $SDRL$ values for two ZIP processes:
$ZIP(0.8,2)$ and $ZIP(0.7,1)$. The shifts in process parameters are
$\phi_{1}=\tau\phi_{0}$ with $\tau\in\\{0.8,0.6\\}$ and
$\lambda_{1}=\delta\lambda_{0}$ with $\delta\in\\{1.2,1.5\\}$. For the first
process, we assume that the size of the Phase I sample is $m=200$ while in the
second $m=500$.

The OOC $ARL$ values are compared for both ZIP-Shewhart charts with either
‘adjusted’ or unadjusted $L$. Also, we provide the OOC $ARL$ values in Case K.
The results show that the use of the ‘adjusted’ $L^{*}$ value results in $ARL$
values that are very close to the theoretical values (i.e. those under Case K)
while there is a reduction in the $SDRL$ values. Note also, that even for
large shifts in process parameters (e.g. for $\delta=1.5$ and $\tau=0.6$) the
respective IC $ARL$ and $SDRL$ in Case U when the unadjusted $L$ is used can
be much different than the corresponding values under Case K.

Table 9: OOC Performance of the ZIP-Shewhart Chart - MLE |  | Case K, $L=5.49$ | Case U, $L=5.49$, $m=200$ | Case U, $L^{*}=4.73$, $m=200$
---|---|---|---|---
$\phi_{1}$ | $\lambda_{1}$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$
0.80 | 2.0 | 301.87 | 301.37 | 936.24 | 2109.12 | 303.87 | 549.85
0.80 | 2.4 | 140.16 | 139.66 | 353.55 | 681.22 | 135.87 | 213.47
0.80 | 3.0 | 59.58 | 59.08 | 121.06 | 196.63 | 55.77 | 75.56
0.64 | 2.0 | 167.70 | 167.20 | 525.50 | 1226.86 | 168.79 | 301.97
0.64 | 2.4 | 77.87 | 77.37 | 197.93 | 382.76 | 75.39 | 119.99
0.64 | 3.0 | 33.10 | 32.60 | 67.44 | 107.50 | 31.17 | 42.15
0.48 | 2.0 | 116.10 | 115.60 | 362.21 | 820.71 | 116.05 | 207.98
0.48 | 2.4 | 53.91 | 53.41 | 138.30 | 283.77 | 52.26 | 81.71
0.48 | 3.0 | 22.92 | 22.41 | 45.54 | 74.05 | 21.58 | 29.25
|  | Case K, $L=5.18$ | Case U, $L=5.18$, $m=500$ | Case U, $L^{*}=4.36$, $m=500$
$\phi_{1}$ | $\lambda_{1}$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$
0.70 | 1.0 | 175.55 | 175.05 | 537.67 | 757.06 | 174.84 | 202.52
0.70 | 1.2 | 98.71 | 98.21 | 260.15 | 351.78 | 97.92 | 109.00
0.70 | 1.5 | 50.78 | 50.27 | 113.41 | 145.43 | 50.17 | 53.25
0.56 | 1.0 | 119.69 | 119.19 | 366.41 | 516.63 | 119.55 | 138.96
0.56 | 1.2 | 67.30 | 66.80 | 177.81 | 241.10 | 66.81 | 73.99
0.56 | 1.5 | 34.62 | 34.12 | 77.84 | 99.37 | 34.22 | 36.19
0.42 | 1.0 | 90.80 | 90.30 | 277.36 | 391.92 | 90.46 | 104.06
0.42 | 1.2 | 51.06 | 50.55 | 134.04 | 180.94 | 50.72 | 56.12
0.42 | 1.5 | 26.27 | 25.76 | 59.01 | 75.39 | 25.93 | 27.24

###  5.2 Out-of-Control Performance of the ZIB-Charts in Case U

In Table 10, we provide the $ARL$ and $SDRL$ values for two ZIB processes:
$ZIB(0.9,0.03,250)$ and $ZIB(0.8,0.01,100)$. Similar to the case of a ZIP
process, the shifts in process parameters are $\phi_{1}=\tau\phi_{0}$ with
$\tau\in\\{0.8,0.6\\}$ and $p_{1}=\delta p_{0}$ with $\delta\in\\{1.2,1.5\\}$.
For the first process, we assume that the size of the Phase I sample is
$m=500$ while in the second $m=1000$.

The results in Table 10 are similar to those in Table 9. Specifically, they
show that the use of the ‘adjusted’ $L^{*}$ gives $ARL$ values that are very
close to the theoretical ones (i.e. those under Case K), while the $SDRL$ are
much lower than the respective ones in the case of ‘unadjusted’ $L$. Like the
case of ZIP processes, the OOC $ARL$ and $SDRL$ can be much larger, compared
to the respective values in Case K, even for large shifts in process
parameters (such as $\delta=1.5$ and $\tau=0.6$) if the ‘unadjusted’ $L$ is
used.

Table 10: OOC Performance of the ZIB-Shewhart Chart - MLE |  |  | Case K, $L=5.09$ | Case U, $L=5.09$, $m=500$ | Case U, $L^{*}=4.73$, $m=500$
---|---|---|---|---|---
$\phi_{1}$ | $p_{1}$ | $n$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$
0.90 | 0.030 | 250 | 248.86 | 248.36 | 483.73 | 833.67 | 250.84 | 382.82
0.90 | 0.036 | 250 | 83.19 | 82.69 | 132.07 | 187.96 | 80.43 | 104.01
0.90 | 0.045 | 250 | 24.68 | 29.17 | 38.55 | 44.62 | 28.41 | 31.08
0.72 | 0.030 | 250 | 88.88 | 88.38 | 173.84 | 306.16 | 89.08 | 135.42
0.72 | 0.036 | 250 | 29.71 | 29.21 | 47.17 | 67.06 | 28.62 | 36.72
0.72 | 0.045 | 250 | 10.60 | 10.09 | 13.78 | 15.67 | 10.12 | 10.76
0.54 | 0.030 | 250 | 54.10 | 53.60 | 105.20 | 180.56 | 54.52 | 83.48
0.54 | 0.036 | 250 | 18.08 | 17.58 | 28.51 | 39.97 | 17.46 | 22.24
0.54 | 0.045 | 250 | 6.45 | 5.93 | 8.36 | 9.35 | 6.17 | 6.38
|  |  | Case K, $L=6.35$ | Case U, $L=6.35$, $m=1000$ | Case U, $L^{*}=5.40$, $m=1000$
$\phi_{1}$ | $p_{1}$ | $n$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$ | $ARL$ | $SDRL$
0.80 | 0.010 | 100 | 272.12 | 271.62 | 848.45 | 1195.33 | 272.55 | 299.14
0.80 | 0.012 | 100 | 152.31 | 151.81 | 410.53 | 556.14 | 153.99 | 161.53
0.80 | 0.015 | 100 | 77.86 | 77.36 | 178.83 | 229.69 | 77.64 | 80.66
0.64 | 0.010 | 100 | 151.18 | 150.68 | 472.46 | 662.73 | 151.42 | 165.80
0.64 | 0.012 | 100 | 84.62 | 84.12 | 229.06 | 309.98 | 89.65 | 90.28
0.64 | 0.015 | 100 | 43.26 | 42.75 | 98.62 | 127.01 | 43.08 | 44.46
0.48 | 0.010 | 100 | 104.66 | 104.16 | 325.56 | 458.49 | 105.11 | 115.89
0.48 | 0.012 | 100 | 58.58 | 58.08 | 158.08 | 213.82 | 58.63 | 62.54
0.48 | 0.015 | 100 | 29.95 | 29.44 | 68.74 | 88.10 | 29.86 | 30.80

##  6 Conclusions

In this work we studied the design and performance of Shewhart-type charts for
zero-inflated processes, when the process parameters are unknown and must be
estimated. Using Monte Carlo simulation and two estimation methods (maximum
likelihood and method of moments) we evaluated the performance of the
considered charts for several Phase I sample sizes, in terms of the
unconditional $ARL$ and $SDRL$ performance measures. The numerical results
showed that in the case of zero-inflated processes, very large Phase I samples
are needed in order to obtain a reliable design of the chart, whose
performance in the case of the estimated parameters is similar to its
theoretical performance. In almost all of the examined cases, the suggested
size $m$ of the Phase I sample is larger than 1000 preliminary observations
while there are cases where even more than 5000 preliminary observations do
not guarantee that the actual performance will be close to the theoretical
one.

For relatively small preliminary samples (e.g. for $m=100$ or 200), the IC
$ARL$ and $SDRL$ values can be very large. Therefore, we do not advise
practitioners to set-up their chart when the process parameters are estimated
from a Phase I sample of this size. Note also that in case of large $\phi_{0}$
values, a small Phase I sample might end up to a sample with only zero values,
making the estimate of process parameters impossible. As a general guideline,
we suggest that Phase I samples with at least 1000 observations are necessary
to obtain reliable estimates of process parameters. As $m$ increases, the IC
$ARL$ and $SDRL$ reduce, but not necessarily in a monotonic way (the former).

In an attempt to assist practitioners in ending up with reliable designs for
their charts, we provided ‘adjusted’ values for the design parameter $L$ in
order to obtain an IC $ARL$ very close to IC $ARL$ in Case K, for a given size
$m$ of the Phase I sample. Note also that the proposed methodology works for
any other nominal $ARL_{0}$ (instead of the IC $ARL$ value in Case K) that
meets practitioners’ needs.

Finally, We used R (see R Core Team (2023)) for all the necessary computations
in this work. The related programs are available from the authors upon
request.

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