# Stein EWMA Control Charts for Count Processes

Christian H. Weiß  Helmut Schmidt University, Department of Mathematics and
Statistics, Hamburg, Germany.Corresponding author. E-Mail<EMAIL_ADDRESS>ORCID: 0000-0001-8739-6631

###### Abstract

The monitoring of serially independent or autocorrelated count processes is
considered, having a Poisson or (negative) binomial marginal distribution
under in-control conditions. Utilizing the corresponding Stein identities,
exponentially weighted moving-average (EWMA) control charts are constructed,
which can be flexibly adapted to uncover zero inflation, over- or
underdispersion. The proposed Stein EWMA charts’ performance is investigated
by simulations, and their usefulness is demonstrated by a real-world data
example from health surveillance.

Key words: attributes data; average run lengths; count time series; EWMA
control charts; Stein identity

##  1 Introduction

The sequential monitoring of count processes
$(X_{t})_{t\in\mathbb{N}=\\{1,2,\ldots\\}}$ (i. e., where the $X_{t}$ have a
quantitative range contained in $\mathbb{N}_{0}=\\{0,1,\ldots\\}$) is of
utmost importance in many application areas, such as the quality control of
manufactured items (counts of defects or non-conformities) or health
surveillance (counts of infections or hospital admissions); see Montgomery
(2009); Weiß (2015, 2018) for examples and references. The primary tool for
managing statistical process control (SPC) for $(X_{t})$ is the use of
attributes control charts. These charts involve plotting specific statistics
sequentially until signals indicate the necessity for corrective action. These
statistics are computed in an online manner from the sequence of incoming
counts $(X_{t})$. The control chart triggers a signal (“alarm”) if the plotted
statistic violates the specified control limits (CLs). The CLs are derived
from the so-called in-control model of $(X_{t})$, i. e., a stochastic model
that assumes $(X_{t})$ to operate under stable conditions. Hence, an alarm (a
violation of the CLs) is interpreted as an indication of a possible process
deterioration (out-of-control situation). Therefore, if the alarm occurs when
the process $(X_{t})$ is genuinely out of control, it is considered a true
alarm. On the other hand, an alarm under in-control conditions is regarded as
a false alarm. It stands to reason that CLs should be chosen to avoid false
alarms for as long as possible, and to receive a true alarm as soon as
possible. The default metric for evaluating these durations until alarm is the
average run length (ARL), i. e., the expected number of plotted statistics
until the first alarm. For a more detailed description of the aforementioned
terms and concepts as well as for further references, the reader may consult
SPC textbooks such as Montgomery (2009); Qiu (2014).

Various control charts for diverse count processes have been developed during
the last decades, covering serially independent or autocorrelated counts, and
counts having full $\mathbb{N}_{0}$ as their range (unbounded counts) or just
the finite subset $\\{0,\ldots,n\\}$ with specified $n\in\mathbb{N}$ (bounded
counts). The most simple type of control chart is obtained by plotting the
counts $X_{t}$ themselves against appropriately chosen CLs, which is called a
c-chart if monitoring unbounded counts and an np-chart for bounded counts.
Such types of Shewhart control chart can be classified as being memory-less as
the $t$th plotted statistic does not comprise information about earlier counts
(at least not beyond the mere effect of autocorrelation). Consequently, while
Shewhart charts might be quick in detecting a sudden strong process shift,
they are slow in detecting small process changes. Therefore, also several
memory-type control charts for counts have been proposed, where in the present
research, our focus is on exponentially weighted moving-average (EWMA) charts.
Some recent references on EWMA-type control charts for counts are Gan (1990);
Borror et al. (1998); Weiß (2011); Rakitzis et al. (2015); Morais et al.
(2018); Morais & Knoth (2020); Anastasopoulou & Rakitzis (2022a, b). Further
references (also on other types of count control charts) can be found in Weiß
(2015, 2018); Alevizakos & Koukouvinos (2020). The default version of the EWMA
chart plots the statistics

$Z_{0}=\mu_{0},\quad Z_{t}\ =\ \lambda\cdot X_{t}\ +\ (1-\lambda)\cdot
Z_{t-1}\quad\text{for }t=1,2,\ldots$ (1)

against a specified lower and upper CL (LCL and UCL, respectively), where
$\mu_{0}>0$ denotes the in-control mean of $(X_{t})$. The smoothing parameter
$\lambda\in(0,1]$ in (1) controls the strength of the memory (the smaller
$\lambda$, the stronger the memory). If $\lambda=1$, then (1) reduces to the
c- or np-chart, respectively.

All the aforementioned count EWMA charts are mainly designed to detect shifts
in the process mean, although they may sometimes (“accidentally”) also react
to increases in variance or changes in the autocorrelation structure. In the
present research, however, our focus is on “more sophisticated” out-of-control
scenarios, namely where the mean changes together with further distributional
properties (i. e., where the process change cannot be traced back to a sole
change in the mean parameter), or where “purely distributional changes” (not
affecting the mean) happen. Such “distributional changes” might be increases
or decreases in dispersion compared to the in-control model (overdispersion or
underdispersion, respectively), or an excessive number of zero counts (zero
inflation), to mention those being most relevant in practice. The basic idea
of our approach is as follows. Many common count distributions can be
characterized by a type of moment identity, referred to as _Stein identity_ ,
which has to hold for a large class of functions (e. g., all bounded functions
on $\mathbb{N}_{0}$). The idea to develop such identities dates back to Stein
(1972, 1986), and further contributions and references can be found in
Sudheesh (2009); Sudheesh & Tibiletti (2012); Landsman & Valdez (2016).
Recently, such Stein identities were successfully used to develop powerful
goodness-of-fit (GoF) tests for counts, see Betsch et al. (2022); Weiß et al.
(2023) for Stein-type GoF-tests for independent and identically distributed
(i. i. d.) counts, and Aleksandrov et al. (2022a, b) for tests for count time
series. Thus, it suggests itself to utilize these Stein identities also for
developing sequential test procedures, namely count control charts for
relevant types of in-control model. In a first research (see Weiß, 2023), this
idea was tried for the special case of i. i. d. Poisson counts, and the
achieved ARL performance was quite appealing. This motivates to develop and
investigate Stein-based control charts for count data on a much broader scale,
namely for various different count distributions and not only for i. i. d. but
also for time series data. More precisely, we focus on the three most common
count distributions in practice, namely Poisson (Poi), negative binomial (NB),
and binomial (Bin), and we include first-order autoregressive (AR$(1)$) models
having either Poi-, NB-, or Bin-distributed marginal distributions in our
research.

The outline of this chapter is as follows. In Section 2, we briefly present
the count models used for this research, and we provide the Stein identities
for the Poi-, NB-, and Bin-distribution. These are used in Section 3 to
construct novel EWMA-type control charts for counts, where the chart design
with respect to diverse out-of-control scenarios is discussed in detail. In
Section 4, results from a simulation study are presented, which allow to
analyze the Stein EWMA charts’ ARL performance. Section 5 then investigates a
real-world data example on registrations in the emergency department of a
children’s hospital, which illustrates the application and interpretation of
the novel Stein EWMA charts in practice. Finally, Section 6 concludes the
article and discusses possible directions for future research.

##  2 Count Models and Stein Identities

The two most common distributions for unbounded counts are the Poi- and NB-
distribution, while the Bin-distribution is the default choice for bounded
counts; see Johnson et al. (2005) for details and properties. The
$\textup{Poi}(\mu)$-distribution with mean $\mu>0$ is known to be
equidispersed, i. e., its variance $\sigma^{2}$ satisfies $\sigma^{2}=\mu$. By
contrast, $\textup{NB}(\nu,\frac{\nu}{\nu+\mu})$ with $\nu,\mu>0$ exhibits
overdispersion relative to the Poi-distribution, because its variance
$\sigma^{2}=(1+\frac{\mu}{\nu})\,\mu$ always exceeds the mean. The
$\textup{Bin}(n,\mu/n)$-distribution with $n\in\mathbb{N}$ and $\mu\in(0,n)$,
in turn, refers to counts having the bounded range $\\{0,\ldots,n\\}$.

There is a huge variety of time series models related to the Poi-, NB-, or
Bin-distribution, see Weiß (2018) for a comprehensive overview. Here, many
models use a _conditional_ Poi-, NB-, or Bin-distribution (regression models)
while the marginal distribution does not belong to any parametric model
family. However, to be able to apply a Stein identity to given data, we need
to specify the corresponding _marginal_ distribution. Regarding the count time
series models proposed so far Weiß (2018), only a considerably smaller number
of models has a Poi-, NB-, or Bin-_marginal_ distribution. For the sake of
studying possible effects of serial dependence on the charts’ performance, we
shall consider three of the AR$(1)$-type processes surveyed by Weiß (2008),
whose model definitions use so-called thinning operators as integer
substitutes of the multiplication:

  * •

the Poi-INAR$(1)$ process (integer AR) defined by

$X_{t}\ =\ \rho\circ X_{t-1}+\epsilon_{t}\quad\text{with i.\,i.\,d.\
}\epsilon_{t}\sim\textup{Poi}\big{(}\mu(1-\rho)\big{)},$ (2)

  * •

the NB-IINAR$(1)$ process (iterated-thinning INAR) defined by

$X_{t}\ =\ \rho\circledast_{\pi}X_{t-1}+\epsilon_{t}\quad\text{with
i.\,i.\,d.\ }\epsilon_{t}\sim\textup{NB}(\nu,\pi)\text{ and
}\pi=\tfrac{\nu}{\mu(1-\rho)+\nu},$ (3)

  * •

the BinAR$(1)$ process defined by

$X_{t}\ =\ \alpha\circ X_{t-1}+\beta\circ(n-X_{t-1})\quad\text{with
}\beta=(1-\rho)\,\tfrac{\mu}{n},\ \alpha=\beta+\rho.$ (4)

Definitions (2) and (4) use the binomial thinning operator, which is defined
by requiring a conditional Bin-distribution, namely $\theta\circ
X|X\sim\textup{Bin}(X,\theta)$ for $\theta\in(0,1)$. Definition (3), in turn,
uses iterated thinning defined as
$\rho\circledast_{\pi}X=\sum_{i=1}^{(\pi\rho)\circ X}Y_{i}$, where the
counting series $(Y_{i})$ is i. i. d. according to $Y_{i}\sim
1+\textup{NB}(1,\pi)$. The crucial point for the subsequent research is the
following:

Models (2)–(4) lead to stationary Markov chains with AR$(1)$-like
autocorrelation function (ACF) $\rho(h)=\rho^{h}$ for time lags
$h\in\mathbb{N}$, where $\rho\in(0,1)$ for all three models. The stationary
marginal distributions are $\textup{Poi}(\mu)$,
$\textup{NB}(\nu,\frac{\nu}{\nu+\mu})$, and $\textup{Bin}(n,\mu/n)$,
respectively, see Weiß (2008). The parameter $\rho$ allows to control the
extent of serial dependence, where the boundary case $\rho\to 0$ leads to i.
i. d. counts.

Independent of the value of $\rho$, we are always concerned with a Poi-, NB-,
or Bin-marginal distribution. Any of these three distributions is uniquely
characterized by a corresponding Stein identity, see Sudheesh & Tibiletti
(2012):

  * •

$X\sim\textup{Poi}(\mu)$ if and only if

$E\big{[}X\,f(X)\big{]}\ =\ \mu\,E\big{[}f(X+1)\big{]}$ (5)

holds for all bounded functions $f:\mathbb{N}_{0}\to\mathbb{R}$;

  * •

$X\sim\textup{NB}(\nu,\frac{\nu}{\nu+\mu})$ if and only if

$(\nu+\mu)\,E\big{[}X\,f(X)\big{]}\ =\ \mu\,E\big{[}(\nu+X)\,f(X+1)\big{]}$
(6)

holds for all bounded functions $f:\mathbb{N}_{0}\to\mathbb{R}$;

  * •

$X\sim\textup{Bin}(n,\mu/n)$ if and only if

$(n-\mu)\,E\big{[}X\,f(X)\big{]}=\mu\,E\big{[}(n-X)\,f(X+1)\big{]}$ (7)

holds for all bounded functions $f:\mathbb{N}_{0}\to\mathbb{R}$.

Identity (5) is commonly referred to as the Stein–Chen identity (see Chen,
1975). Note that the identities (5)–(7) constitute non-trivial statements only
if $f$ is not constant on $\mathbb{N}_{0}$, and if $f$ is not identical to
zero on $\mathbb{N}$.

In the subsequent Section 3, we shall derive control charts from the
identities (5)–(7). As these identities have to hold for all bounded functions
$f$ under in-control conditions, we can select any choice of $f$ for defining
the control charts. This degree of freedom can be used to achieve particular
sensitivity regarding a specified type of out-of-control scenario.

##  3 Stein EWMA Charts for Counts

The three identities (5)–(7) always depend on three types of moment: the mean
$\mu$, the moment $E\big{[}X\,f(X)\big{]}$, and a moment involving $f(X+1)$.
The idea of the GoF-tests in Aleksandrov et al. (2022a, b); Weiß et al. (2023)
as well as of the Poisson EWMA charts in Weiß (2023) was to derive a statistic
by solving the identities in a certain way, and to substitute the involved
population moments by appropriate types of sample moments. In Weiß (2023), two
types of constructing an EWMA control chart were analyzed, and it turned out
that one of these types is clearly superior, namely the one called “ABC-EWMA
chart”. Building on this experience, we propose the following Stein EWMA
charts for sequentially monitoring Poi-, NB-, or Bin-counts, respectively,
where $E_{0}[\cdot]$ expresses that the expectation is computed with respect
to the in-control model. For all three Stein EWMA charts, we compute

$\begin{array}[]{@{}l}A_{0}=E_{0}\big{[}X\,f(X)\big{]},\qquad A_{t}\ =\
\lambda\cdot X_{t}\,f(X_{t})\ +\ (1-\lambda)\cdot A_{t-1},\\\\[4.30554pt]
C_{0}=\mu_{0},\qquad C_{t}\ =\ \lambda\cdot X_{t}\ +\ (1-\lambda)\cdot
C_{t-1},\quad\text{for }t=1,2,\ldots\end{array}$ (8)

Furthermore, we compute for the

  * •

$\textup{Poi}(\mu_{0})$ in-control model:

$\begin{array}[]{@{}l@{\quad}l}B_{0}=E_{0}\big{[}f(X+1)\big{]},&B_{t}\ =\
\lambda\cdot f(X_{t}+1)\ +\ (1-\lambda)\cdot B_{t-1},\\\\[4.30554pt]
Z_{0}^{\textup{S}}=1,&\displaystyle Z_{t}^{\textup{S}}\ =\
\frac{A_{t}}{B_{t}\,C_{t}},\qquad\text{for }t=1,2,\ldots;\end{array}$ (9)

  * •

$\textup{NB}(\nu,\frac{\nu}{\nu+\mu_{0}})$ in-control model:

$\begin{array}[]{@{}l@{\quad}l}\lx@intercol
<EMAIL_ADDRESS>\lx@intercol B_{t}\ =\ \lambda\cdot(\nu+X_{t})\,f(X_{t}+1)\ +\
(1-\lambda)\cdot<EMAIL_ADDRESS>Z_{0}^{\textup{S}}=1,&\displaystyle Z_{t}^{\textup{S}}\ =\
\frac{(\nu+C_{t})\,A_{t}}{B_{t}\,C_{t}},\qquad\text{for
}t=1,2,\ldots;\end{array}$ (10)

  * •

$\textup{Bin}(n,\mu_{0}/n)$ in-control model:

$\begin{array}[]{@{}l@{\quad}l}\lx@intercol
<EMAIL_ADDRESS>\lx@intercol B_{t}\ =\ \lambda\cdot(n-X_{t})\,f(X_{t}+1)\ +\ (1-\lambda)\cdot
<EMAIL_ADDRESS>Z_{0}^{\textup{S}}=1,&\displaystyle
Z_{t}^{\textup{S}}\ =\ \frac{(n-C_{t})\,A_{t}}{B_{t}\,C_{t}},\qquad\text{for
}t=1,2,\ldots\end{array}$ (11)

We define the respective _Stein EWMA chart_ by plotting the statistics
$Z_{t}^{\textup{S}}$ against appropriately chosen
$\textup{LCL}<1<\textup{UCL}$, where we expect $Z_{t}^{\textup{S}}$ to vary
closely around 1 under in-control assumptions.

The actual chart design now comprises two steps. First, we have to select the
function $f$ involved in (8)–(11). Here, one could generally choose any
bounded function on $\mathbb{N}_{0}$, recall (5)–(7), except trivial choices
like $f$ being constant on $\mathbb{N}_{0}$, or $f\equiv 0$ on $\mathbb{N}$.
But not any choice of $f$ will lead to an appealing chart performance.
Instead, $f$ has to be chosen with respect to the anticipated out-of-control
scenario, in an analogous way as proposed by Aleksandrov et al. (2022a, b);
Weiß et al. (2023) and Weiß (2023). The idea is to interpret $f$ as a weight
function within the “A- and B-moments”, which puts unequal weight on the
integers in $\mathbb{N}_{0}$. For a given anticipated out-of-control scenario,
one should put most weight on those regions of $\mathbb{N}_{0}$ where one gets
the strongest departures from the in-control model. If we want to uncover zero
inflation, for example, we should choose an $f$ with relatively large weight
close to zero counts, whereas for “general” overdispersion (i. e., if the
probability mass function (PMF) is flattened compared to the in-control
model), it is advantageous to put more weight on large counts. So in
accordance to the recommendations in Weiß (2023), the following choices of $f$
are considered in these two cases:

  * •

for uncovering overdispersion relative to the in-control model, we use the
linear weights $f(x)=|x-1|$;

  * •

for uncovering zero inflation relative to the in-control model, we use the
root weights $f(x)=|x-1|^{1/4}$.

The case of underdispersion has not been investigated by Weiß (2023). Thus, as
a starting point, we follow the findings of Weiß et al. (2023) on GoF-tests
and

  * •

for uncovering underdispersion relative to the in-control model, we use the
inverse weights $f(x)=1/(x+1)$.

The described approach for choosing the weight function is later illustrated
in some more detail when discussing Figure 1. The ARL performance of the
aforementioned choices of $f$ is analyzed in Section 4 below. As we shall see,
the underdispersion scenario is much more demanding than overdispersion or
zero inflation. Therefore, further choices for $f$ shall be considered later
in Section 4.2.

After having specified $f$, the second step of chart design is the choice of
the triple $(\lambda,\textup{LCL},\textup{UCL})$. Assume for the moment that
the smoothing parameter $\lambda$ has already been fixed. Then, LCL and UCL
are determined based on ARL considerations. While different ARL concepts exist
in the literature (Knoth, 2006), it is common to use the zero-state ARL for
evaluating the in-control performance. The most simple solution (the one used
here) are symmetric CLs of the form $\textup{LCL}=1-L$ and $\textup{UCL}=1+L$,
where $L$ is chosen such that the desired in-control ARL (ARL0) is met in
close approximation (the textbook choice is the target value $370$).
Alternatively, one could define asymmetric CLs by also considering the out-of-
control ARL performance, but this is only reasonable if a particular out-of-
control scenario has been fixed. For example, if one assumes that the out-of-
control models differ from the in-control one solely in terms of the mean
$\mu$, then asymmetric CLs may allow to obtain an unbiased ARL performance
with respect to $\mu$, i. e., the ARL as a function of $\mu$ is maximal in the
in-control mean $\mu_{0}$ and decreases symmetrically around $\mu_{0}$, see
Section 4 in Morais & Knoth (2020). But as we shall consider a broad variety
of out-of-control scenarios, we restrict to symmetric CLs here, i. e.,
$\mu_{0}\mp L$ for ordinary EWMA and $1\mp L$ for Stein EWMA charts.

At this point, let us recall that the above approach for choosing $f$ assumes
that we have specified a _single_ relevant out-of-control scenario, such as
“zero inflation”. If the application context allows for _various_ out-of-
control scenarios (e. g., if underdispersion could also be possible), one can
run multiple Stein EWMA charts in parallel, each designed for a different type
of deterioration. This is later done in Section 5 when monitoring the
emergency counts. There, one Stein EWMA chart is designed to detect
overdispersion, another one for zero inflation, and a third one for
underdispersion. Then, the observed pattern of alarms enables a kind of
targeted diagnosis. For example, if the “zero inflation”-chart is the first to
trigger an alarm, we conclude that we might be confronted with zero inflation
(rather than with underdispersion etc.). Certainly, if running many charts in
parallel, the risk increases that one of these charts gives a false alarm
(“multiple testing”). Generally, the risk of false alarms is controlled by
setting a target value ARL0 for the in-control ARL. As described before, in
our simulation study, we set ARL${}_{0}\approx 370$ for each single chart for
comparability. But if multiple control charts are applied simultaneously, it
would also be possible to determine their CLs such that the _joint_ in-control
ARL meets a target value, i. e., the ARL would be defined as the expected time
until one of the charts triggers the first alarm.

Another remark refers to the choice of the smoothing parameter $\lambda$. To
ensure a sufficient memory, one often chooses small values for $\lambda$, such
as $\lambda\in\\{0.25,0.10,0.05\\}$. The actual choice of $\lambda$ might be
done based on out-of-control ARLs. For example, we could first compute several
candidate triples $(\lambda_{i},\textup{LCL}_{i},\textup{UCL}_{i})$ having the
same ARL0. Then, we fix a specified out-of-control scenario and select that
triple where the corresponding out-of-control ARL is closest to a given target
value. In this way, one might end up with different $\lambda$ for different
Stein EWMA charts. But as we choose $f(x)$ with respect to different out-of-
control scenarios, comparable chart designs are most easily obtained by a
unique $\lambda$. Furthermore, all Stein EWMA charts follow the same type of
recursive scheme, which makes it plausible to use a unique value for
$\lambda$. Thus, in the sequel, we follow the practice in Weiß (2023) and set
$\lambda=0.10$ throughout our analyses.

Let us conclude this section with a note on out-of-control ARLs. As mentioned
before, there are a couple of competing ARL concepts for evaluating a control
chart’s out-of-control performance, see Knoth (2006), which are typically
defined with respect to different positions of the change point. Here, the
change point $\tau$ expresses the time where the process turns out of control,
i. e., the process is in control (out of control) for $t<\tau$ ($t\geq\tau$).
The out-of-control zero-state ARL assumes that the process change happens
right at the beginning of process monitoring, i. e., the change point equals
$\tau=1$. The conditional expected delay CED$(\tau)$, by contrast, assumes a
later change point $\tau>1$, where the limit $\tau\to\infty$ leads to the
steady-state ARL. While $\tau=1$ would happen for a misspecified in-control
model, late change points are often more realistic in practice (but one does
not know the true value of $\tau$ in advance). Fortunately, it turned out in
the initial analyses of Weiß (2023) that the exact position of $\tau$ does not
have a substantial effect on the computed out-of-control ARL values, i. e.,
the Stein EWMA charts showed roughly the same performance for early and late
process changes. Since the charts of the present study are defined in complete
analogy to those of Weiß (2023), we expect the value of the zero-state ARL to
be representative also for $\tau>1$ (for “problematic” types of control chart,
see the discussion in Knoth et al. (2023)). For this reason, we restrict our
subsequent analyses to zero-state ARLs, and these are computed based on
simulations. So one simulates the considered process for $R$ times (we use
$R=10^{4}$), applies the specified control chart to it, and determines the
time until the first alarm. In this way, one gets $R$ run lengths
$l_{1},\ldots,l_{R}$, and the sample mean thereof provides an approximate
value of the actual ARL.

##  4 Simulation-based Performance Analyses

###  4.1 Overdispersion or Zero Inflation

Let us start our performance analyses with the case of overdispersion or zero
inflation (compared to the in-control model). Strictly speaking, a zero-
inflated distribution also exhibits increased dispersion, but this increase is
caused by a single point mass in zero. With overdispersion, by contrast, we
refer to a flattened PMF compared to the in-control model’s PMF. For unbounded
counts, we use the NB-distribution for generating overdispersion, and the
zero-inflated Poisson (ZIP) distribution for zero inflation. In the bounded-
counts case, we use the beta-binomial (BB) distribution for overdispersion and
the zero-inflated binomial (ZIB) distribution for zero inflation, see Appendix
A in Weiß (2018) for details on these distributions. For the sake of a unique
representation, we specify any of the aforementioned models by the mean $\mu$
and by an appropriate dispersion index, namely by
$I_{\textup{P}}=\sigma^{2}/\mu$ for unbounded counts and
$I_{\textup{B}}=n\sigma^{2}/\big{(}\mu(n-\mu)\big{)}$ for bounded counts. Note
that $I_{\textup{P}}=1$ for the Poi-distribution while $I_{\textup{P}}>1$ for
NB and ZIP. Analogously, $I_{\textup{B}}=1$ for the Bin-distribution and
$I_{\textup{B}}>1$ for BB and ZIB. The in-control means are set at either
$\mu_{0}=2$ (low counts) or $\mu_{0}=5$ (medium counts). The data-generating
process (DGP) produces either i. i. d. counts ($\rho=0$) or AR$(1)$-like
counts with $\rho=0.5$, recall (2)–(4). The simulated ARLs rely on $10^{4}$
replications, the target value for ARL0 is $370$.

Table 1: Simulated ARLs of ordinary EWMA and Stein EWMA chart for Bin-counts
with $n=10$. BB and ZIB with $I_{\textup{B}}=5/3$.

$\mu_{0}$ | $\mu=$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$
---|---|---|---|---|---|---|---|---|---|---
(a) i. i. d. counts
|  | EWMA |  | 0.7805 | EWMAS | $|x-1|$ | 0.534 | EWMAS | $|x-1|^{1/4}$ | 0.4235
2 | ZIB | 69.2 | 87.9 | 51.6 | 22.7 | 26.1 | 29.2 | 16.6 | 19.1 | 21.9
| Bin | 171.5 | 370.2 | 99.3 | 240.2 | 369.5 | 550.6 | 191.5 | 370.6 | 671.0
| BB | 71.5 | 90.0 | 51.8 | 25.9 | 29.2 | 33.3 | 30.8 | 40.5 | 55.6
|  | EWMA |  | 0.974 | EWMAS | $|x-1|$ | 0.2115 | EWMAS | $|x-1|^{1/4}$ | 0.0511
5 | ZIB | 69.2 | 87.9 | 51.6 | 18.9 | 19.9 | 20.6 | 12.9 | 14.0 | 15.5
| Bin | 162.5 | 369.5 | 164.1 | 307.2 | 370.1 | 443.4 | 311.5 | 369.5 | 417.2
| BB | 66.7 | 88.2 | 66.3 | 25.1 | 26.9 | 29.0 | 27.4 | 28.9 | 30.3
(b) AR$(1)$ counts with $\rho=0.5$
|  | EWMA |  | 1.191 | EWMAS | $|x-1|$ | 0.639 | EWMAS | $|x-1|^{1/4}$ | 0.568
2 | ZIB | 77.9 | 73.8 | 55.8 | 22.9 | 24.3 | 25.3 | 23.8 | 26.9 | 30.0
| Bin | 384.8 | 370.1 | 158.0 | 247.1 | 369.7 | 554.4 | 211.9 | 371.2 | 634.3
| BB | 105.0 | 105.7 | 77.2 | 33.0 | 39.5 | 47.0 | 44.7 | 58.9 | 77.1
|  | EWMA |  | 1.493 | EWMAS | $|x-1|$ | 0.225 | EWMAS | $|x-1|^{1/4}$ | 0.0528
5 | ZIB | 30.8 | 29.9 | 28.0 | 7.9 | 7.7 | 7.3 | 7.1 | 6.9 | 6.5
| Bin | 258.0 | 369.1 | 257.3 | 316.7 | 370.9 | 424.1 | 293.9 | 370.9 | 421.8
| BB | 86.3 | 96.1 | 85.9 | 35.3 | 37.6 | 39.5 | 37.9 | 39.4 | 40.7
Notes: | In-control ARL printed in bold font, CL $L$ shown in italic font.
| “EWMA” = ordinary EWMA; “EWMAS” = Stein EWMA,
| where weight functions $f(x)=|x-1|$ and $|x-1|^{1/4}$.
Interpre- |  |  |  |  | pure |  | zero inflation
tation: | sole mean change |  | distrib. |  |  |  |
|  |  |  |  | change |  | overdispersion

Table 2: Simulated ARLs of ordinary EWMA and Stein EWMA chart for NB-counts
with $I_{\textup{P}}=5/3$. ZIP with $I_{\textup{P}}=5/3$, and overdispersed NB
(“oNB”) with $I_{\textup{P}}=5/2$.

$\mu_{0}$ | $\mu=$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$
---|---|---|---|---|---|---|---|---|---|---
(a) i. i. d. counts
|  | EWMA |  | 1.156 | EWMAS | $|x-1|$ | 0.349 | EWMAS | $|x-1|^{1/4}$ | 0.3146
2 | ZIP | 506.6 | 462.0 | 149.7 | 139.4 | 257.2 | 481.5 | 54.2 | 81.0 | 124.9
| NB | 605.8 | 370.7 | 133.1 | 172.1 | 370.9 | 892.2 | 154.0 | 369.9 | 1001.2
| oNB | 171.9 | 135.1 | 75.6 | 45.2 | 67.2 | 103.8 | 52.2 | 86.9 | 163.2
|  | EWMA |  | 1.805 | EWMAS | $|x-1|$ | 0.1554 | EWMAS | $|x-1|^{1/4}$ | 0.0883
5 | ZIP | 342.6 | 407.5 | 261.9 | 116.2 | 143.9 | 170.2 | 22.7 | 24.7 | 26.8
| NB | 444.7 | 370.8 | 205.3 | 267.3 | 369.8 | 522.7 | 260.0 | 370.1 | 537.2
| oNB | 130.2 | 124.7 | 94.8 | 42.9 | 51.3 | 59.1 | 55.5 | 68.7 | 87.3
(b) AR$(1)$ counts with $\rho=0.5$
|  | EWMA |  | 1.855 | EWMAS | $|x-1|$ | 0.45 | EWMAS | $|x-1|^{1/4}$ | 0.4415
2 | ZIP | 1287.7 | 505.7 | 238.0 | 108.3 | 167.0 | 266.3 | 90.6 | 139.7 | 219.4
| NB | 770.9 | 369.7 | 200.4 | 187.4 | 370.7 | 710.6 | 182.7 | 370.7 | 768.8
| oNB | 288.9 | 178.8 | 115.2 | 61.1 | 93.5 | 141.8 | 75.4 | 123.1 | 210.1
|  | EWMA |  | 2.78 | EWMAS | $|x-1|$ | 0.177 | EWMAS | $|x-1|^{1/4}$ | 0.1105
5 | ZIP | 502.7 | 408.0 | 284.2 | 147.8 | 178.4 | 214.4 | 93.6 | 110.2 | 131.1
| NB | 510.8 | 369.6 | 242.5 | 278.0 | 370.8 | 494.2 | 276.7 | 370.2 | 492.8
| oNB | 183.8 | 156.4 | 124.4 | 60.1 | 71.6 | 83.0 | 78.5 | 97.1 | 122.0
Notes: | In-control ARL printed in bold font, CL $L$ shown in italic font.
| “EWMA” = ordinary EWMA; “EWMAS” = Stein EWMA,
| where weight functions $f(x)=|x-1|$ and $|x-1|^{1/4}$.
Interpre- |  |  |  |  | pure |  | zero inflation
tation: | sole mean change |  | distrib. |  |  |  |
|  |  |  |  | change |  | overdispersion

As the case of overdispersion or zero inflation relative to an in-control Poi-
distribution was already analyzed by Weiß (2023), we restrict our subsequent
analyses to in-control NB- and Bin-models, see Tables 1–2 for the obtained
results. The Bin-results are quite close to the Poi-results anyway, which is
not surprising as these distributions are related to each other by the Poisson
limit theorem. Let us start our discussion with the Bin-case in Table 1. The
design of this table (and of any further table) is as follows. It consists of
twelve $3\times 3$-blocks with a negative (positive) mean shift on the left
(right) and no mean shift in the center column. In the Bin-rows, the binomial
distribution is preserved, whereas we have distributional changes in the ZIB-
and BB-rows. Note that at the bottom of each table, there is a scheme for
interpreting the ARLs within each block. Part (a) of each table refers to i.
i. d. counts, part (b) to AR$(1)$-like counts. Note that the BB- and ZIB-
AR$(1)$ model are constructed as in (3.27) of Weiß (2018) by replacing the
Bin-thinnings in (4) by BB- or ZIB-thinnings, respectively. Finally, the first
column of blocks always refers to the ordinary EWMA chart (1), whereas the
remaining two columns refer to Stein EWMA charts, i. e., to (11) in case of
Table 1.

Part (a) of Table 1 just confirms the findings of Weiß (2023) for the Poi-
case. If we are concerned with the textbook situation, i. e., if there is a
change solely in the mean parameter (the distribution is preserved otherwise),
then the ordinary EWMA chart is clearly the best choice. But if there is an
additional distributional change, or even only a distributional change while
the mean is kept fixed, then the Stein EWMA charts become superior. More
precisely, if we are concerned with overdispersion (BB-model), then the weight
function $f(x)=|x-1|$ leads to the lowest ARLs, whereas zero inflation is best
detected using $f(x)=|x-1|^{1/4}$. It should be noted that we draw the same
conclusions from part (a) of Table 2, which refers to the NB-case (in-control
level of dispersion is $I_{\textup{P}}=5/3$) with Stein EWMA chart (10).
There, overdispersion (relative to the in-control model) was generated by an
NB-distribution with higher dispersion, namely $I_{\textup{P}}=5/2$, whereas
we used the ZIP-distribution with $I_{\textup{P}}=5/3$ for causing zero
inflation. Thus, we generally conclude that if monitoring i. i. d. counts and
if being concerned with overdispersion under out-of-control conditions, the
Stein EWMA chart with $f(x)=|x-1|$ is the best choice, whereas
$f(x)=|x-1|^{1/4}$ performs best under zero inflation. If the type of out-of-
control situation is not clear in advance, it is recommended to run all three
charts of Tables 1–2 simultaneously. As these charts show a rather different
ARL performance, they can be used for a kind of targeted diagnosis, i. e., one
can conclude from the observed pattern of alarms on the type of out-of-control
situation. For example, if the chart with $f(x)=|x-1|^{1/4}$ signals first, we
conclude on zero inflation.

Next, let us look at parts (b) of Tables 1–2, i. e., on the case of positively
correlated counts. Note that in Table 2, the overdispersed NB-counts are still
generated by the NB-IINAR$(1)$ model (3), whereas the ZIP-INAR$(1)$ model was
used for zero inflation, i. e., model (2) with ZIP-innovations
$(\epsilon_{t})$. In most cases in parts (b), CLs have to be increased and the
charts’ ARL performances get worse. But it still holds that the Stein EWMA
charts are superior under overdispersion or zero inflation. As the only
difference, the ARL performances of both Stein EWMA charts are now quite
similar under zero inflation in the Bin-case, but not for the NB-case. So
altogether, the conclusions done for i. i. d. counts still apply, but with
generally increased out-of-control ARLs due to the serial dependence.

(a)$x$ (b)$x$

Figure 1: PMF plots for Good with $I_{\textup{P}}=0.5$ vs. Poi, for (a)
$\mu_{0}=2$ and (b) $\mu_{0}=5$. Dotted line proportional to $1/(x+1)$.

###  4.2 Underdispersion

Underdispersion is the opposite phenomenon to overdispersion, i. e., the PMF
is concentrated more closely around the mean. While there are hardly any
models for underdispersion of bounded counts, the underdispersion of unbounded
counts received more interest in the literature. For the subsequent
simulations, we use the Good distribution (see Weiß, 2018, Appendix A) for
generating underdispersion relative to the Poi-distribution (see Table 3),
whereas for underdispersion relative to an NB-distribution, we use either an
NB-distribution with less dispersion or the Poi-distribution (see Table 4). It
quickly gets clear from Tables 3–4 that underdispersion is quite demanding for
process monitoring. If looking first at the ordinary EWMA chart (first columns
of blocks), we recognize severely increasing ARLs in the presence of
underdispersion. This is reasonable as fluctuations of the DGP are reduced
such that small mean shifts (we have $\pm 0.25$) hardly lead to a violation of
the CLs. The second columns in Tables 3–4 refer to the Stein EWMA charts (9)
and (10), where we used $f(x)=1/(x+1)$ as recommended by Weiß et al. (2023) in
the context of GoF-tests. While this chart indeed performs well for the low
mean $\mu=2$, the ARL performance is very bad for $\mu=5$, again demonstrating
that process monitoring in the presence of underdispersion is quite demanding.
To better understand these difficulties and to find a solution, let us look at
Figure 1, where we visualize the effect of underdispersion (black) compared to
the in-control model (grey) by PMF plots. It can be seen that the strongest
deviations are in a small area below $\mu_{0}$. For $\mu_{0}=2$ in (a), the
main deviations happen for $x\in\\{0,1,2\\}$. In fact, $f(x)=1/(x+1)$
(symbolized by the dotted graphs in Figure 1) indeed puts most weight to this
region, which explains the good ARL performance in this case. For $\mu_{0}=5$
in (b), however, the main deviations are for $x\in\\{2,\ldots,5\\}$. As
$f(x)=1/(x+1)$ puts relatively low weight into this region, we now get a poor
ARL performance.

Table 3: Simulated ARLs of ordinary EWMA and Stein EWMA chart for Poi-counts.
Good with $I_{\textup{P}}=3/4$ (first row) and $I_{\textup{P}}=1/2$ (second
row).

$\mu_{0}$ | $\mu=$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$
---|---|---|---|---|---|---|---|---|---|---
(a) i. i. d. counts
|  | EWMA |  | 0.877 | EWMAS | $1/(x+1)$ | 0.223 | EWMAS | $p_{\textup{P}}(x+2)$ | 0.608
2 | Poi | 252.6 | 369.1 | 106.1 | 274.6 | 368.9 | 470.8 | 538.9 | 370.3 | 271.7
| Good | 622.4 | 948.9 | 158.5 | 96.9 | 142.3 | 249.5 | 90.7 | 71.4 | 60.5
| Good | 3611.8 | 6346.9 | 380.6 | 32.3 | 42.2 | 63.0 | 29.1 | 26.2 | 24.3
|  | EWMA |  | 1.388 | EWMAS | $1/(x+1)$ | 0.1775 | EWMAS | $p_{\textup{P}}(x+2)$ | 0.293
5 | Poi | 309.9 | 371.4 | 185.1 | 352.9 | 370.5 | 398.1 | 526.1 | 368.7 | 268.9
| Good | 1006.0 | 1015.9 | 327.2 | $>10^{4}$ | $>10^{4}$ | $>10^{4}$ | 228.4 | 149.3 | 106.4
| Good | 8154.5 | 7882.4 | 1168.9 | $>10^{4}$ | $>10^{4}$ | $>10^{4}$ | 52.5 | 40.1 | 33.2
(b) AR$(1)$ counts with $\rho=0.5$
|  | EWMA |  | 1.351 | EWMAS | $1/(x+1)$ | 0.2467 | EWMAS | $p_{\textup{P}}(x+2)$ | 0.7235
2 | Poi | 627.2 | 371.0 | 162.0 | 274.8 | 370.0 | 478.4 | 530.0 | 370.5 | 273.2
| Good | 2177.5 | 814.6 | 261.3 | 153.7 | 242.0 | 439.2 | 143.5 | 108.9 | 89.6
| Good | $>10^{4}$ | 2866.0 | 581.6 | 47.6 | 62.4 | 92.3 | 43.7 | 36.7 | 33.2
|  | EWMA |  | 2.123 | EWMAS | $1/(x+1)$ | 0.1707 | EWMAS | $p_{\textup{P}}(x+2)$ | 0.345
5 | Poi | 432.2 | 369.8 | 236.3 | 332.3 | 370.1 | 395.3 | 514.5 | 370.1 | 280.7
| Good | 1369.3 | 909.2 | 446.1 | 2652.2 | 4029.5 | 5159.4 | 226.6 | 161.8 | 122.0
| Good | $>10^{4}$ | 4264.3 | 1402.0 | 1856.0 | 6087.1 | $>10^{4}$ | 69.8 | 53.9 | 44.3
Notes: | In-control ARL printed in bold font, CL $L$ shown in italic font.
| “EWMA” = ordinary EWMA; “EWMAS” = Stein EWMA,
| where weight functions $f(x)=1/(x+1)$ and $p_{\textup{P}}(x+2)$.
Interpre- | sole mean change |  | pure |  |  |  |
tation: |  |  |  |  | distrib. |  | increasing
|  |  |  |  | change |  | underdispersion

Table 4: Simulated ARLs of ordinary EWMA and Stein EWMA chart for NB-counts
with $I_{\textup{P}}=5/3$. Underdispersed NB (“uNB”) with
$I_{\textup{P}}=4/3$.

$\mu_{0}$ | $\mu=$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$ | $\mu_{0}-0.25$ | $\mu_{0}$ | $\mu_{0}+0.25$
---|---|---|---|---|---|---|---|---|---|---
(a) i. i. d. counts
|  | EWMA |  | 1.156 | EWMAS | $1/(x+1)$ | 0.2215 | EWMAS | $p_{\textup{N}}(x+2)$ | 0.4163
2 | NB | 605.8 | 370.7 | 133.1 | 240.3 | 371.5 | 590.6 | 367.7 | 370.3 | 325.2
| uNB | 1531.2 | 799.1 | 202.7 | 280.0 | 380.5 | 568.9 | 315.1 | 213.5 | 152.7
| Poi | 5998.4 | 3237.7 | 429.8 | 106.1 | 128.8 | 179.9 | 93.7 | 70.8 | 57.3
|  | EWMA |  | 1.805 | EWMAS | $1/(x+1)$ | 0.165 | EWMAS | $p_{\textup{N}}(x+2)$ | 0.22
5 | NB | 444.7 | 370.8 | 205.3 | 301.9 | 369.1 | 452.9 | 399.3 | 369.1 | 328.4
| uNB | 1072.6 | 840.9 | 372.4 | 1185.9 | 1531.8 | 1921.6 | 455.4 | 313.1 | 226.8
| Poi | 4236.6 | 3623.7 | 986.3 | 7006.1 | $>10^{4}$ | $>10^{4}$ | 132.6 | 96.9 | 76.5
(b) AR$(1)$ counts with $\rho=0.5$
|  | EWMA |  | 1.855 | EWMAS | $1/(x+1)$ | 0.2412 | EWMAS | $p_{\textup{N}}(x+2)$ | 0.4626
2 | NB | 770.9 | 369.7 | 200.4 | 266.3 | 369.8 | 514.0 | 440.5 | 369.8 | 293.9
| uNB | 1797.1 | 685.4 | 315.4 | 233.3 | 319.1 | 441.7 | 248.6 | 177.5 | 137.2
| Poi | $>10^{4}$ | 2329.1 | 713.9 | 103.7 | 136.9 | 190.5 | 87.6 | 68.6 | 58.2
|  | EWMA |  | 2.78 | EWMAS | $1/(x+1)$ | 0.1727 | EWMAS | $p_{\textup{N}}(x+2)$ | 0.247
5 | NB | 510.8 | 369.6 | 242.5 | 308.7 | 370.7 | 442.5 | 414.8 | 370.0 | 306.6
| uNB | 1148.0 | 700.0 | 420.9 | 787.5 | 973.5 | 1222.5 | 329.4 | 241.2 | 191.5
| Poi | 4918.5 | 2317.8 | 1082.8 | 1689.4 | 2741.1 | 4019.6 | 124.0 | 96.6 | 78.9
Notes: | In-control ARL printed in bold font, CL $L$ shown in italic font.
| “EWMA” = ordinary EWMA; “EWMAS” = Stein EWMA,
| where weight functions $f(x)=1/(x+1)$ and $p_{\textup{N}}(x+2)$.
Interpre- | sole mean change |  | pure |  |  |  |
tation: |  |  |  |  | distrib. |  | increasing
|  |  |  |  | change |  | underdispersion

To get a good performance regarding underdispersion, Figure 1 suggests to put
most weight somewhat below the in-control mean, which leads to the following
idea. The in-control PMF itself (plotted in grey in Figure 1), interpreted as
a weight function, puts most weight around the mean of the in-control
distribution. Thus, we may just shift this PMF downwards to move most weight
below the mean. Inspired by Figure 1 and some simulation experiments, the idea
was developed to define $f(x)$ by simply shifting the in-control PMF by $2$.
More precisely, for the in-control Poi-model of Table 3, we use
$p_{\textup{P}}(x+2)$ as the weight function, and $p_{\textup{N}}(x+2)$ for
the NB-model of Table 4. Here, $p_{\textup{P}}(\cdot)$ and
$p_{\textup{N}}(\cdot)$ abbreviate the PMFs of the in-control Poi- and NB-
distribution, respectively. The computed chart designs and ARLs are summarized
in the respective third column of Tables 3–4. We now get notably decreasing
ARLs for increasing underdispersion in any scenario, both under i. i. d. and
AR$(1)$-like counts.

##  5 An Illustrative Data Application

Weiß & Testik (2015) analyzed a large set of daily count time series referring
to registrations in the emergency department of a children’s hospital. More
precisely, these emergency counts were determined per 5-min interval from
08:00:00 to 23:59:59 on a day (so length $T=192$), and the full set of time
series covers the period from February 13 to August 13, 2009. Weiß & Testik
(2015) used the sixteen time series from February 13 to 28 to develop the in-
control model (Phase-I analysis), namely a Poi-INAR$(1)$ model with in-control
mean $\mu_{0}=2.1$ and dependence parameter $\rho_{0}=0.78$. Control charts
based on this model were then applied to prospective process monitoring
(Phase-II application). While most Phase-II series did not contradict the in-
control model, Weiß & Testik (2015) recognized a few unusual days, two of
which shall now serve as illustrative data examples.

Let us start our analyses by designing the control charts based on the in-
control model. To make the results consistent with Section 4, the CLs of each
chart are chosen such that the in-control ARL is close to 370. On the one
hand, we consider the c-chart and the ordinary EWMA chart (1) as competitors.
On the other hand, the new Stein EWMA chart (9) was used together with four
weight function: $f(x)=|x-1|$ and $|x-1|^{1/4}$ for indicating possible
overdispersion or zero inflation, and $f(x)=1/(x+1)$ and $p_{\textup{P}}(x+2)$
for underdispersion. While the ARLs of the c-chart are computed numerically
exactly by using the Markov-chain approach (see Weiß, 2018, Section 8.2.2), we
again used simulations with $10^{4}$ replications for the remaining charts.
Here, the c-chart is most difficult to design due to discreteness; the best
design has $\textup{LCL}=0$ and $\textup{UCL}=6$ (so an only one-sided design,
where counts $X_{t}>6$ cause an alarm) and ARL${}_{0}\approx 326.2$. The EWMA
chart designs (again with $\lambda=0.1$) are truly two-sided, namely

  * •

EWMA (1) with $L=1.851$ and ARL${}_{0}\approx 370.3$;

  * •

Stein EWMA (9) with $f(x)=|x-1|$, $L=0.848$, ARL${}_{0}\approx 370.5$;

  * •

Stein EWMA (9) with $f(x)=|x-1|^{1/4}$, $L=0.829$, ARL${}_{0}\approx 370.5$;

  * •

Stein EWMA (9) with $f(x)=1/(x+1)$, $L=0.2994$, ARL${}_{0}\approx 370.5$;

  * •

Stein EWMA (9) with $f(x)=p_{\textup{P}}(x+2)$, $L=0.9594$, ARL${}_{0}\approx
370.2$.

(a) c-chart: | (b) Ordinary EWMA chart:
---|---
$t$ |  $t$
(c) Stein EWMA with $f(x)=|x-1|$: |  (d) Stein EWMA with $f(x)=|x-1|^{1/4}$:
$t$ |  $t$
(e) Stein EWMA with $f(x)=1/(x+1)$: |  (f) Stein EWMA with $f(x)=p_{\textup{P}}(x+2)$:
$t$ |  $t$
Figure 2: Emergency counts from March 28, 2009, see Section 5: c-chart in (a),
ordinary EWMA chart in (b), and different Stein EWMA charts in (c)–(f). CLs as
dashed lines, solid center line, and first alarm at dotted line.

As a first illustrative example, we apply these control charts to the
emergency counts collected on March 28, 2009, see Figure 2. The c-chart
triggers an alarm at $t=23$, the ordinary EWMA only rather late at $t=171$,
both indicating that the in-control model is violated. The situation gets more
clear if looking at the four Stein EWMA charts in Figure 2. The charts in (c)
and (d) both trigger a very early alarm at $t=6$, whereas those of (e) and (f)
do not trigger an alarm at all. This indicates that we are confronted with
overdispersion and zero inflation. In fact, comparing the sample properties of
the emergency series from March 28, 2009, to the in-control model, we note a
slight increase in the mean $\mu$ (from 2.1 to $\approx 2.323$) and a
substantial increase in dispersion $I_{\textup{P}}$ (from 1 to $\approx
1.312$). Furthermore, the number of zeros equals 27, being larger than 23.5 as
expected under the in-control model. So the Stein EWMA charts gave a clear
diagnosis of the type of out-of-control situation. In addition, their alarm at
$t=6$ was not only much faster than those of c-chart and ordinary EWMA chart,
but also than those of the control charts in Weiß & Testik (2015) (these
trigger at $t=48$ and $t=50$, respectively).

(a) c-chart: | (b) Ordinary EWMA chart:
---|---
$t$ |  $t$
(c) Stein EWMA with $f(x)=|x-1|$: |  (d) Stein EWMA with $f(x)=|x-1|^{1/4}$:
$t$ |  $t$
(e) Stein EWMA with $f(x)=1/(x+1)$: |  (f) Stein EWMA with $f(x)=p_{\textup{P}}(x+2)$:
$t$ |  $t$
Figure 3: Emergency counts from July 16, 2009, see Section 5: c-chart in (a),
ordinary EWMA chart in (b), and different Stein EWMA charts in (c)–(f). CLs as
dashed lines, solid center line, and first alarm at dotted line.

As the second illustrative example, see Figure 3, we consider the emergency
counts from July 16, 2009, where no results are reported by Weiß & Testik
(2015). This time, neither c-chart nor ordinary EWMA chart trigger an alarm,
so the user would conclude that the emergency series is in control. Looking at
the Stein EWMA charts, however, the ones in (e) and (f) signal at times $t=35$
and $t=74$, respectively, so we seem to be confronted with underdispersion. In
fact, the sample value of $I_{\textup{P}}$ is $\approx 0.710$ being notably
smaller than 1, but also the mean has decreased from 2.1 to $\approx 1.911$.
As we know from Section 4.2, the ordinary EWMA chart is hardly able to detect
mean shifts in the presence of underdispersion, so the novel Stein EWMA charts
constitute a welcome complement for this scenario.

##  6 Conclusions and Future Research

For the monitoring of either Poi-, NB-, or Bin-counts, corresponding Stein
EWMA charts were proposed, which are constructed by utilizing the respective
Stein identity. Their ARL performance was investigated by simulations, both
for i. i. d. and AR$(1)$-like counts. It turned out that overdispersion is
best detected by using the linear weights $f(x)=|x-1|$, and zero inflation by
the root weights $f(x)=|x-1|^{1/4}$. Count monitoring in the presence of
underdispersion, however, turned out to be quite demanding. While the ordinary
EWMA charts performs very poorly in this case, the Stein EWMA chart with
inverse weights $f(x)=1/(x+1)$ has appealing out-of-control ARLs for low
counts. Even more promising is the weight function obtained by a downward
shift of the in-control PMF. These findings were also confirmed by the data
example on emergency counts. Nevertheless, it appears that the monitoring of
underdispersion requires additional research activity.

There are further directions for future research. As cumulative sum (CUSUM)
charts often show a better out-of-control performance than EWMA charts, it
would be relevant to develop and investigate Stein CUSUM charts for count
processes. Here, a residuals-based approach in analogy to Weiß & Testik (2015)
would be attractive, as this would not only be applicable to Poi-, NB-, or
Bin-marginals, but also to in-control models having a Poi-, NB-, or Bin-
conditional distribution. Furthermore, Stein EWMA or CUSUM charts would also
be interesting for continuously distributed variables data; for this case,
Stein identities can be found in Sudheesh (2009); Landsman & Valdez (2016),
among others.

#### Acknowledgments

The author thanks the two referees for their useful comments on an earlier
draft of this article. This research was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer
437270842.

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