1. INTRODUCTION
Control chart has long been playing a central role as a statistical process control (SPC) method for detecting an abnormal change in manufacturing processes. It is well known that the traditional Shewhart X control chart reacts slowly to small shifts of the process mean. Page (1954) introduced the cumulative sum (CUSUM) control chart to improve the performance when monitoring small to moderate shifts. For more details on control charts, we refer to Montgomery (2009). Although sampling is usually conducted with a fixed sampling interval (FSI) to maintain control charts, the variable sampling interval (VSI) is also adopted to improve the efficiency of monitoring: see Reynolds et al. (1988, 1990) and Saccucci et al. (1992). Epprecht et al. (2010) recently applied the VSI control scheme to the exponential weighted moving average (EWMA) chart, which is an alternative to the Shewhart control chart designed for faster monitoring. In the VSI control scheme, sampling intervals are planned to be varying according to the information obtained from the precedent control statistic and the performance is usually evaluated by the average time to signal (ATS) triggering an outofcontrol condition. For more details, we refer to Reynolds et al. (1990).
When dealing with the rare events such as the infection rate of rare diseases or the number of nonconforming units in a highquality manufacturing process, traditional control charts for attribute often lose their efficiency owing to the excessive number of zeros in count data. Those zeros cause an overdispersion for the data, and thereby, underestimating the target values in the monitoring process: see Woodall and Mohammed (2006). In order to overcome this defect, several authors considered control charts adopting zeroinflated models for count data. Sim and Lim (2008) considered Shewharttype control charts for zeroinflated Poisson (ZIP) and binomial (ZIB) distributions based on the Jeffrey’s prior intervals and Blyth Still intervals. Wang (2009) introduced an improved p control chart using adjusted confidence intervals to cope with the case of small nonconforming rates. He et al. (2012) and Rakitzis et al. (2016) recently studied the CUSUM control chart for zeroinflated Poisson and binomial distributions. Conventionally, the detection of an increase, rather than a decrease, in model parameters has been a core issue, because the increase is mainly due to the process deterioration induced by an assignable cause.
In this study, we consider the VSI CUSUM control chart considered by Reynolds et al. (1990), focusing on the ZIB distribution proposed by Hall (2000). We particularly deal with the task of monitoring an increase of the proportion parameter in ZIB processes using the VSI CUSUM chart and compare its performance with that of the FSI CUSUM chart considered by Rakitzis et al. (2016) via measuring their ATS triggering an outofcontrol condition. Moreover, we put our efforts to develop an R package, attrCUSUM, for an easy implementation of the attribute VSI CUSUM control chart, which is available from the comprehensive R archive network (CRAN).
This paper is organized as follows. Section 2 introduces zeroinflated count models and summarizes their probabilistic properties. Section 3 presents the FSI and VSI CUSUM control charts in ZIB processes. Section 4 demonstrates the effects of the VSI CUSUM chart in ZIB processes through numerical experiments. Section 5 describes the R package attrCUSUM by illustrating some examples using ZIB, Poisson and negative binomial distributions. Section 6 provides concluding remarks.
2. THE ZEROINFLATED MODEL FOR COUNT DATA
The zeroinflated model for count data has been popular in analyzing excessive number of zeros in observations: see, for example, Lambert (1992), Hall (2000), and Vieira et al. (2000). Let X be a random variable (r.v.) for zeroinflated count data. Then, the probability mass function (pmf) of X is expressed as follows (Agarwal et al., 2002):
where I(x= 0) denotes the Dirac distribution on x= 0, W is a nonnegative r.v. with pmf p_{w}(x;θ)with parameter vector θ and the ρ ∈[0,1] is the zeroinflation parameter. When ρ= 0 the distribution of X coincides with that of W whereas when ρ= 1 the distribution of X reduces to the Dirac distribution on x= 0 From equation (1), the cumulative distribution function (cdf) of X is obtained as:
where F_{w}(x:θ) is the cdf of W with parameter vector θ. Further, it can be easily seen that the kth moments and the variance of X are given as:
Thus, the dispersion of X is given as:
where D_{w}= V(W)/EW denotes the dispersion of W This particularly shows that the existence of frequent zeros causes the overdispersion.
The maximum likelihood estimator (MLE) $({\widehat{\rho}}^{ml},\hspace{0.17em}\hspace{0.17em}{\widehat{\theta}}^{ml})$ of parameters (ρ,θ) given x_{1},…,x_{m}, is obtained as the maximizer of the loglikelihood function:
The MLE can be calculated using a numerical maximization procedure easily accessible from statistical programming languages such as R (R Core Team, 2016). Since we here focus on Phase II control chart, we assume that the parameter values of ρ and θ are known or have been estimated precisely from a precedent large Phase I sample.
In this study, we consider the VSI CUSUM control chart based on the ZIB distribution. We do this because the ZIB process is the most appropriate when we deal with count data with an excessive number of zeros and the sampled data is viewed as a group of size n>1 (Noorossana et al., 2011). Assume that X follows a ZIB distribution with parameters (ρ, n, p) Then, its pmf p_{x}(x;ρ, n, p) is given as (Hall, 2000):
where (n, p)∈(N, [0,1]) is the parameters of a standard binomial distribution. The mean, variance and dispersion of X are given by:
Given observations x_{1},,x_{m} from a ZIB process with parameters (ρ, n, p) where n is assumed to be known, the MLE $\text{o}\left({\widehat{\rho}}^{ml},{\widehat{\rho}}^{ml}\right)$ of the parameters (ρ, p) is obtained by solving the nonlinear equations (Vieira et al., 2000):
where x is the sample mean and ${\overline{x}}^{+}={\displaystyle {\sum}_{i=1}^{m}{x}_{t}I\left({x}_{t}>0\right)/{\displaystyle {\sum}_{t=1}^{m}I\left({x}_{t}>0\right)}}$ is the mean of the positive values of the observations. Since the MLE $\left({\widehat{\rho}}^{ml},{\widehat{\rho}}^{ml}\right)$ has no closed form, a numerical optimization procedure should be hired in its calculation.
3. FSI AND VSI CUSUM CONTROL SCHEME FOR ZIB PROCESS
In this section, we describe the FSI and VSI CUSUM control charts with ZIB processes, including the Markov chain approach proposed by Brook and Evans (1972).
3.1 CUSUM Control Statistic for ZIB Process
We consider the monitoring process for the proportion p of nonconforming units of ZIB processes with parameters (ρ, n, p) where the ρ is assumed to be known and to remain unchanged and the sample size n is fixed as in Rakitzis et al. (2016), Noorossana et al. (2011), and Sim and Lim (2008). We assume that X_{i},i=1, 2,…, are independent and identically distributed ZIB (ρ ,n,δ p_{0}) r.v.s, where δ > 0 is a positive constant that determines the magnitude of a shift in p_{0} Notice that when δ =1 the process is regarded as incontrol, whereas it is regarded as outofcontrol when δ ≠1. As mentioned by Rakitzis et al. (2016), when δ >1 the outofcontrol condition is attributed to an increase in the proportion of nonconforming units, mainly caused by a deterioration in the process performance. Meanwhile,δ <1 implies that there is an improvement in the process, which can be used to set up a potential future direction for process adjustment.
In this study, we design the onesided VSI CUSUM control chart for monitoring the proportion increase in ZIB processes, based on their corresponding FSI CUSUM charts. Traditional onesided FSI CUSUM control charts for detecting a mean increase are expressed as the plot of CUSUM statistic:
where ${\tilde{c}}_{0}\in [0,h)$ is a starting value, usually set to be 0, k ≥ μ_{x} is a reference value, which has a roll to inhibit the CUSUM statistic from drifting toward the control limit h CUSUM charts get less sensitive to small shifts but more sensitive to larger shifts as k increases. Other starting values are sometimes used, for instance, when considering a fast initial response (FIR) feature (Lucas and Crosier, 1982). The process is regarded as outofcontrol when the signal, ${\tilde{c}}_{1}\ge h,$ is triggered
In the VSI CUSUM control scheme, a modified statistic is usually used. The VSI upper onesided CUSUM control statistic for detecting a mean increase is expressed as follows (Reynolds et al., 1990):
where c_{0} ∈[−k, h) is a starting value. As mentioned by Reynolds et al. (1990), a remarkable difference between ${\tilde{C}}_{t}$ in (2) and C_{t} in (3) lies in whether or not they record the negative CUSUM values: ${\tilde{C}}_{t}$ resets any negative CUSUM values to zero, whereas C_{t} records negative CUSUM values and proceeds its accumulation from zero for the next sample. Like the conventional FSI CUSUM chart, the VSI CUSUM chart signals a mean increase whenever C_{t}≥h It can be easily seen that C_{t}≥h is equivalent to ${\tilde{C}}_{t}\ge h.$ For this reason, we employ the modified CUSUM statistic for both FSI and VSI CUSUM charts.
The VSI CUSUM chart is operated with two sampling intervals 0<d_{s}≥d_{l} where d_{s} denotes a shorter sampling interval, required to be as short as possible, and d_{l} is a longer one which should be properly chosen for the process to run without sampling. Given a warning limit w∈(−k, h) the sampling interval is determined by the current value of the CUSUM statistic. When the process is seemingly close to outofcontrol state, namely w≤C_{t}<h the next sample is taken after a relatively shorter time, d_{s} for reducing the delay time until an assignable cause is detected; otherwise, namely if C_{t}<w, the next sample is taken after a relatively longer time, d_{l} for preserving the predetermined average sampling frequency (see Figure 1). When d_{s}=d_{l} the VSI scheme can be regarded as a FSI scheme. For convenience, it is assumed that d_{s}=d_{l}= 1 when considering the FSI scheme in this study. The condition implies that the d_{s}=d_{l}=1 ATS is the same as the average number of sample to signal (ANSS)s.
The VSI CUSUM statistic C_{t} which uses negative values, provides more warning limits than the conventional FSI CUSUM statistic ${\tilde{C}}_{t}$ Thus, it provides more diverse design parameter candidates and yields a better parameter design. This is the main reason to use the modified statistic C_{t} instead of ${\tilde{C}}_{t}$ .
Assume that one is interested in the quick detection of a shift δ >1 in p_{0} The performance of the FSI CUSUM chart is usually measured by the ANSS in an outofcontrol condition.
Let (ρ, n, δp_{0}, c_{0}, k, h) denote the ANSS for the case of the starting value, reference value, and control limit c_{0}, k, h in ZIB processes with parameters (ρ, n, δp_{0}). Then, the optimal pair (k^{*}, h^{*}) is obtained by solving the optimization problem stated below:
subject to
$\text{ANSS}\left(\rho ,n,{p}_{0},{c}_{0}{k}^{\ast},{h}^{\ast}\right)={\text{ANSS}}_{0},$
ANSS(ρ, n, p_{0}, c_{0}, k, h) = ANSS_{0} ,
where the ANSS_{0} denotes a predetermined incontrol ANSS. Given ANSS_{0} the procedure of finding an optimal pair (k^{*}, h^{*}) consists of the following two steps (Rakitzis et al., 2016): i) finding all possible pairs satisfying the condition ANSS(ρ, n, p_{0} ,c_{0} ,k, h) = ANSS_{0} ii) selecting the one satisfying (4) among such pairs.
Let ψ_{s}, ψ_{l} be the average number of samples before the occurrence of an outofcontrol signal, except the initial state, that use d_{s} and d_{l} respectively, and let ρ_{s} be the longrun proportion of shorter sampling intervals d_{s}.Then, the following relations hold (Reynolds et al., 1990):
$\text{ANSS}=1+{\psi}_{s}+{\psi}_{l},$
From the above, we see that ATS = d. ANSS, where d= d_{s}p_{s}+d_{l}(1ρ_{s}) Note that d is regarded as the average sampling interval.
Similarly to the FSI CUSUM chart, we define ATS (ρ, n, δp_{0}, c_{0}, k, h, w, d_{s}, d_{l}) to be the ATS for the case of the warning limit, shorter interval and longer sampling interval w, d_{s}, d_{l}: the others are the same as in the FSI CUSUM chart. Then, given d_{s} the optimal pair k^{*}, h^{*}, w^{*} is obtained by solving the optimization problem given below:
subject to
${\text{ATS}}_{0}:=\text{ATS}\left(\rho ,\hspace{0.17em}n,\hspace{0.17em}{p}_{0},\hspace{0.17em}{c}_{0},\hspace{0.17em}{k}^{*},\hspace{0.17em}{h}^{*},\hspace{0.17em}{w}^{*},\hspace{0.17em}{d}_{s},\hspace{0.17em}{d}_{l}\right)={\text{ANSS}}_{0}.$
In this study, the procedure of finding an optimal (k^{*}, h^{*}, w^{*}) is based on the optimal pair (k^{*}, h^{*}) found in the FSI CUSUM scheme for procedural efficiency and practicality, which implies that both FSI and VSI chart have the same ANSS function (Reynolds et al., 1988). Given ANSS_{0}, d_{s} and (k^{*}, h^{*}) in the FSI CUSUM scheme, the procedure of finding an optimal warning limit w^{*} consists of the two steps: i) for all possible warning limits, calculate d_{l} such that ATS_{0}= ANSS_{0}, ii) for all pairs (w, d_{l}) select the one satisfying (7). The details on the calculations of ANSS and ATS and obtaining d_{l} values based on the Markov chain approach are described in Subsection 3.2.
3.2 The Markov Chain Approach for CUSUM Control Chart
The optimization problems in (4) and (7) can be solved approximately based on the discretization of state space and the Markov chain approach proposed by Brook and Evans (1972). For details on the Markov chain approach in the VSI CUSUM scheme, we also refer to Reynolds et al. (1990). For describing the procedure of discretization and the Markov chain approach in this study, we first consider an interval [L, H with L, H> 0 This interval is then divided into (H/d)+(L/d) subintervals with length satisfying 0 < d < H, satisfying H/d ∈ N, L / d ∈N, such as:
Suppose that there is a discretetime Markov chain consisting of states −L, −L + d,…, H − d, H with 0 < d < H, H/d ∈N and L/d∈N, where −L, −L + d,…, H − d denote transient states and H denotes an absorbing state. The corresponding transition probability matrix P and the transient probability matrix Q of size (H/d) + (L/d), (H/d) + (L/d) are given as follows:
where q_{i,j}= P(C_{t}= jC_{t1}= i), i,j=L, L+d,…,Hd, 0=(0,0,…,0)^{T} and 1 = (1,1,…,1)^{T}. Note that each q_{i,j} can be approximated as stated below:
Define the initial probability vector α for transient state with C_{0}=c_{0} as follows:
where 0 < d < H, H / d ∈N and L / d ∈N. Similarly, each α_{j} is calculated approximately by:
Let μ= (μ_{L}, μ_{L+d},…,μ_{Hd})^{T} be the solution of the linear equation (IQ)·μ= 1 where I denotes the identity matrix and 1 =(1,1,…,1)^{T}. Each μ_{i}, i = L, L + d,…, Hd is interpreted as the average number of steps before the Markov chain is absorbed when it starts from transient state i Then, the ANSS (ρ, n, δp_{0}, c_{0} , k, h) is computed by:
Let μ_{s} be the solution of the linear equation (IQ). μ_{s}= v_{s} where v_{s}= (v_{s,L}, v_{s, L+d},…,v_{s,Hd})^{T} with
Then, it can be easily seen that ψ_{s} is obtained by ψ_{s} = α^{T}μ_{s} and the d_{1} satisfying ATS_{0} = ANSS_{0} can be obtained using equation (5) and the equation: 1 = d_{s}ρ_{s}+ d_{l}(1ρ_{s}) with δ = 1. Therefore, the ATS is obtained by equation (6).
In the evaluation of ANSS and ATS, we keep an accuracy to two decimal places as in Rakitzis et al. (2016). To be more specific, we use the settings as follows:
Notice that a better approximation result can be obtained as mentioned by Rakitzis et al. (2016), but such a setting can induce the dimension of the transient matrix Q to increase geometrically, giving rise to some cumbersome issues as to memory allocation, computation time, and the validity of related matrices.
4. EFFECTS OF THE VSI CUSUM CONTROL SCHEME IN ZIB PROCESS
In this section, we illustrate an example of the VSI CUSUM control scheme in ZIB processes. The performance of the FSI CUSUM and VSI CUSUM charts is compared by their ATS values. For the design of the FSI ZIBCUSUM chart, we refer to Rakitzis et al. (2016). They provide optimal parameters in the quick detection of shift 1.2 in p_{0} by conducting considerably many numerical experiments for several cases with starting value c_{0} and ANSS_{0} = 370.4. Using the ANSS_{0} of the FSI CUSUM chart, we compute the ATS values of FSI and VSI CUSUM charts with ATS_{0} = ANSS_{0} corresponding to the optimal warning limit in the quick detection of a shift 1.2 in p_{0} when p_{0}=0 and d_{s} The ATS value in the FSI CUSUM chart is regarded as the ANSS value. The average sampling frequency (ASF) values, that is the ratio of the ANSS to the ATS, are also presented for comparison. The results are provided in Figure 2 and Tables 13, wherein ρ_{s}_{,0} denotes the longrun proportion of shorter sampling intervals when the process is incontrol. From these results, we can see that i) In the detection of the shift 1.2 in ρ_{0} (which is the most interesting case), for all cases, there is a reduction in ATS when adopting the VSI control scheme. Moreover, smaller values of d_{s} imply smaller ATS values when all other parameters are fixed (see also Figure 2). ii) There is no guarantee that for all possible shifts, the optimal warning limit w yields smaller ATS values than the ANSS of the FSI chart. When δ is large (i.e. the large shift in p_{0} for some cases, such as ZIB (0.9,100,δ ⋅0.02) and δ=3, the ATS value of the VSI ZIBCUSUM chart becomes larger than the ANSS of the FSI ZIBCUSUM chart, which might be due to the longer sampling interval. However, this is not an important case because the CUSUM control chart is mainly designed for monitoring small to moderate shifts. In fact, one may raise a question as to an optimal d_{s} since the smaller d_{s} is, the smaller outofcontrol ATS becomes, which is a common phenomenon in VSI CUSUM charts. Although the shortest possible d_{s} is required, it must be at least greater than or equal to the minimum time required for taking a sample and the corresponding longer sampling time d_{l} must be admissible to allow the process of interest to run without taking a sample. For a more detailed discussion, we refer to Reynolds et al. (1990). Table 2
Although we mainly illustrate the ZIB case, the VSI control chart also works properly and efficiently in detecting assignable causes for count processes: see Subsection 5.2 for example. The VSI CUSUM control scheme is somewhat cumbersome in application compared to the FSI control chart, but it provides attractive opportunities to practitioners. The attribute VSI CUSUM chart is easily implemented by the R package attrCUSUM described in Section 5.
5. SOFTWARE
In this section, we develop the R package attrCUSUM for an easy use of the VSI CUSUM chart for count data in practice. The attrCUSUM deals with the Markov chain approach for control charts, which is available from the CRAN. The latest version of the package is available from the authors upon request. This package utilizes the Markov chain approach by Brook and Evans (1972) (see also Reynolds et al. (1990) for details). The core computational algorithms use the R package RcppArmadillo by Eddelbuettel and Sanderson (2014), which is for the use of the Armadillo C++ library by Sanderson (2010). The Armadillo is a C++ template library for high quality linear algebras, aiming at good balance between usability and computation speed. The attrCUSUM provides useful information in the design of VSI CUSUM control charts for (zero inflated) binomial/Poisson processes, and has merit of easy extension to other count processes.
5.1 Examples of Usage with Zero Inflated Binomial Process
The function getAve_zibinom in the attrCUSUM provides useful information on the design of the upper onesided CUSUM control chart for ZIB processes. In fact, it is a wrapper function of getAve and can be easily extended to any count processes. It also gives an object of list including ANSS, ATS and transition probability matrices, and, further, provides the function getContl_zibinom, a wrapper function of getContl, which returns the most suitable control limits h for a given ANSS_{0} value. Below, we provide some example.:
First of all, the following package should be installed and attached:
Here, we only illustrate the case of ZIB(0.9, 200, 0.01) with c_{0} = 0, w = 0 and k = 0.47. Assume that X_{i} is a ZIB process with parameters (0.9, 200, δ⋅0.01). For reproducing the result from Rakitzis et al. (2016) and applying the VSI control scheme, we find a suitable control limit h with the incontrol ANSS being near 370.4 as stated below:

> getContl_zibinom(rho = 0.9,

+ size = 200,

+ prob = 0.01,

+ anss.target = 370.4,

+ refv = 0.47,

+ c.zero = 0)

$refv.act

[1] 0.47

$c.zero.act

[1] 0

$sol1

contl1 ANSS1

6.5300 370.3765

$sol2

contl2 ANSS2

6.5400 389.5988
We find that the suitable control limit h is given as 6.53 (notice that in Rakitzis et al. (2016), the CUSUM statistic signals the outofcontrol condition when ${\tilde{C}}_{t}>h$ > h but not ${\tilde{C}}_{t}\ge h$ ).
We compute d_{1} of the CUSUM control chart when k = 0.47, h = 6.53, c_{0} = 0, w = 0 and 0.1 d_{s} = as stated below:

> res0 < getAve_zibinom(rho = 0.9,

+ size = 200,

+ prob = 0.01,

+ refv = 0.47,

+ refv = 0.47,

+ c.zero = 0,

+ warnl = 0,

+ ds = 0.1)

> res0$ANSS

[1] 370.3765

> res0$dl

[1] 1.516956
The outof control ANSS and ATS of ZIB process with (0.9, 200, 0.01(1.2)) and w = 0 are obtained as follows:

> res1 < getAve_zibinom(rho = 0.9,

+ size = 200,

+ prob = 0.01×1.2,

+ refv = 0.47,

+ contl = 6.53,

+ contl = 6.53,

+ c.zero = 0,

+ warnl = 0,

+ ds = 0.1,

+ dl = res0$dl)

> res1$ANSS

[1] 183.0429

> res1$ATS

[1] 172.8257
Note that invalid arguments will end up with an error message:

> res0 < getAve_zibinom(rho = 1,

+ size = 1,

+ prob = 0.01,

+ refv = 0.47,

+ contl = 6.53,

+ c.zero = 0,

+ warnl = 0,

+ ds = 0.1)
Error:
CheckArgs_dzibinom(x, rho, size, prob, log)

1: Argument ‘rho’ must be a numeric value in [0,1)

2: Argument ‘size’ must be a positive integer value
5.2 Other Count Models
It is well known that a great number of nonconformities per sample situations can be modeled by a Poisson distribution, so that the Poisson CUSUM chart for nonconformities becomes an unquestionably popular procedure: see, for example, Lucas (1985) and White et al. (1997).
Let, X_{i}, i 1,2,…, be the number of nonconformities observed in consecutive random samples of constant size from a production process of interest. We assume that X_{i} ’s are independent and identically distributed Poisson r.v.s with mean $\lambda +\delta \sqrt{\lambda}.$ . In this case, the VSI CUSUM control chart can be well designed by using the function getAve_pois and getContl_pois similarly to the case of ZIB processes.
Suppose that one is interested in detecting the shift of 0.2 from the mean in units of standard deviation, that is δ = 0.2. An example of the ATS and ASF values of the VSI CUSUM chart with ATS_{0} = ANSS_{0} = 370.4, when the data is Poisson with mean 4 +δ ⋅ 2, can be found in Figure 3 and Table 4. Notice that the optimal (k^{*}, h^{*}) is obtained through considerable amount of numerical experiments as stated in Subsection 3.1. Although the optimal warning limit, in the view of the ATS value, is found to be 4.20, we also consider other warning limits because in case the longer sampling intervals , d_{l} corresponding to the optimal warning value, are too long, it might not be acceptable for practitioners. The results in Table 4 reveal that there are desirable reductions in terms of ATS, confirming the validity of the VSI CUSUM control scheme.
Note that the function getAve and getContl can be easily extended to other count models. For instance, assume that one wishes to compute the ANSS when the data is obtained from a negative binomial distribution of size 2 and success probability 0.5. For k = 4.5 and c_{0} = 0, a suitable control limit h, allowing the incontrol ANSS value to be around 400, can be obtained as stated below:

> getContl(anss.target = 400,

+ refv = 4.5,

+ c.zero = 0,

+ process = function(x) {

+ dnbinom(x, size = 2, prob = 0.5)

+ })

$refv.act

[1] 4.5

$c.zero.act

[1] 0
Then, the corresponding longer sampling interval d_{l} for (k, h) = (4.5, 7.1), c_{0} = 0, d_{s} = 0.1 and w = −2 is given as follows:

> res0 < getAve(refv = 4.5,

+ contl = 7.1,

+ c.zero = 0,

+ process = function(x) {

+ dnbinom(x,

+ size = 2,

+ prob = 0.5)

+ },

+ warnl = 2,

+ ds = 0.1)

> res0$dl

[1] 1.522315
The outof control ANSS and ATS values when the size is equal 2.5 are also obtained as follows:

> res1 < getAve(refv = 4.5,

+ contl = 7.1,

+ .zero = 0,

+ process = function(x) {

+ dnbinom(x,

+ size = 2.5,

+ prob = 0.5)

+ },

+ warnl = 2,

+ ds = 0.1,

+ dl = res0$dl)

> res1$ANSS

[1] 164.7614

> res1$ATS

[1] 135.5315
6. CONCLUDING REMARKS
In this study, we considered the VSI CUSUM chart proposed by Reynold et al. (1990), particularly focusing on its application to the monitoring a small to moderate increase of the proportion parameter in ZIB distributions proposed by Hall (2000). We employed the Markov chain approach taken by Brook and Evans (1972) for the calculation of ATS values and compared its performance with that of the FSI CUSUM chart studied by Rakitzis et al. (2016). Our findings in numerical experiments show that the VSI CUSUM chart is superior to the FSI CUSUM chart when monitoring an increase of the proportion parameter in ZIB processes in terms of ATS. We also developed an R package attrCUSUM to implement the VSI CUSUM chart more efficiently, which has merit of an easy extension to other count processes of interest. The method proposed in this study can be extended to more sophisticated CUSUM charts such as the twosided VSI CUSUM chart, which is left as our future project.