Alternative to Detecting Changes in the Mean of an Autoregressive Fractionally Integrated Process with Exponential White Noise Running on the Modified EWMA Control Chart

Abstract

The modified exponentially weighted moving-average (modified EWMA) control chart is an improvement on the traditional EWMA control chart. Herein, we provide more details about the modified EWMA control chart using various values of an additional design parameter for detecting small-to-moderate shifts in the process mean of an autoregressive fractionally integrated (ARFI(p, d)) process with exponential white noise running thereon. The statistical performances of the two charts were evaluated in terms of the average run length (ARL) obtained by solving integral equations (IEs). This provides an exact formula with proven existence and uniqueness verified by applying Banach’s fixed-point theorem. The accuracy of the proposed formula for the ARL was compared with the ARL derived by using the numerical IE technique for the out-of-control state. Although their accuracies were identical for various out-of-control situations and long-term memory processes, the exact formula method required less than 0.01 s to compute the ARL whereas the numerical IE method took 3–4 s. The strengths of using the exact formula are that it is simple to calculate and the computational time is significantly reduced. Comparing their standard deviations of the run length and median run lengths yielded the same results. Finally, practical examples with real-life data corresponding to ARFI(p, d) processes with exponential white noise are provided to demonstrate the applicability of the proposed exact formula.

1. Introduction

One of the primary goals of statistical process control (SPC) is the sensitive identification of an out-of-control situation and the differentiation between two distinct sources of variation in a production process: common and assignable causes. The former is attributable to the inherent variability of a process whereas the latter is attributable to unknown factors that are not inherent in the process. A process operating with only common-cause variation is in control whereas a process with special-cause variation is out of control. Control charts are a vital component of the SPC toolkit and are effective at identifying the assignable cause and special-cause variations. Indeed, it is a highly effective method for identifying and eliminating infrequent process deviations (cf., Montgomery, [1]).

A control chart is a graphical representation of the upper control limit (UCL), the center line (CL), and the lower control limit (LCL). When the plotted statistic falls inside the control limits, the process is stable and in control. Any point that falls outside the control limits indicates that the process is out of control. Therefore, the necessary corrective steps must be implemented to bring the process back into the in-control state. There are two types of control charts: those with no memory and those with memory. The most well-known memoryless control chart is the Shewhart control chart [1], which is based solely on the most current observations and historical data is disregarded. For this reason, it is most effective at detecting large shifts in the process parameter of interest. Numerous studies have been aimed at enhancing the sensitivity and detection capability of the exponentially weighted moving-average (EWMA) control chart [2,3,4,5,6]. The modified EWMA control chart developed by Patel and Divecha [7] is an improvement thereon. Khan et al. [8] presented the modified EWMA control chart in greater depth and published a generalized version of it. Gan [9] introduced three modified EWMA statistics for detecting small shifts in a process parameter, for which the Shewhart control chart is less effective. Patel and Divecha [7] presented the modified EWMA control chart that is effective at detecting small and abrupt shifts in a process parameter. Khan et al. [10] recently developed the modified EWMA statistic to improve its ability to rapidly detect process parameter shifts by including a second parameter. As opposed to the traditional EWMA statistic, the modified one is capable of detecting shifts in the overall mean. Schmid [11] subsequently suggested the modified EWMA control chart for monitoring processes involving time-series data. All of these authors concluded that the modified EWMA control chart is more capable of detecting shifts in a process parameter than the standard EWMA control chart; for more details, refer to ([12,13]).

Herein, we propose a new EWMA control chart using the modified EWMA statistic that is efficient at detecting changes in the mean of underlying time-series processes. Time-series processes involving wind speed, air temperature, air quality, and hydrological data have hyperbolic autocorrelation functions that decrease slowly and are referred to as long-memory processes. Autoregressive fractionally integrated moving-average (ARFIMA(p, d, q)) models with the fractional differencing parameter d $(d\in (0,\hspace{0.17em}\hspace{0.17em}0.5))$ are widely used to represent long-range dependent behavior [14,15,16,17]. The ARFIMA model can be simplified to the ARFI model by imposing p, d, q = 0, Although many types of control charts have been used for time-series modeling—see for instance, [18,19,20])—our focus is on detecting changes in the mean of a long-memory ARFI process running on the modified EWMA control chart.

Numerous researchers have investigated the construction of forecasting models using serially correlated observations that have included the modified EWMA control chart. In a time series, the observations can be generated from a seemingly random process. However, especially involving econometric observations, there can be both autoregressive (AR) and/or moving-average (MA) components. Another important thing to consider when building a model is how to measure the errors. Smaller errors indicate a more efficient model. Commonly, normally distributed white noise signifies errors in a time-series model involving autocorrelated data. However, under certain circumstances, white noise can be exponentially distributed. For example, Jacob and Lewis [21] modeled an ordered ARMA(1,1) process by considering that the observations are exponential white noise. Ibazizen and Fellag [22] used Bayesian concepts for an AR(1) process with exponential white noise, while Pereira and Turkman [23] employed exponential white noise in the Bayesian analysis of threshold AR models. Suparman [24] recently developed estimators for the AR model parameters with exponential white noise for cases where the order is unknown.

One-sided problems are important in many fields, such as engineering (loading capacity, tear strength, etc.), environmental science (concentration of particulate matter, ozone level, high tide, etc.), economics, and finance (interest rate, unemployment rate, riskiness of financial assets, etc.), for which we are interested only in the increase (or only in the decrease) in the parameter of interest. The dynamics can be evaluated using the one-sided modified EWMA control chart.

The average run length (ARL) is an important metric used to evaluate the performances of control charts. The in-control ARL (ARL₀) is defined as the average number of observations (or monitoring points) before a signal point falls outside the control limits while assuming that the process is in control. In contrast, the out-of-control ARL (ARL₁) is the average number of observations that are required before a shift in the mean is detected when the process is out of control: ARL₀ should be as large as possible while ARL₁ should be as small as possible. There are several techniques for computing the ARL, such as Monte Carlo simulation, Markov chain, Martingale, and numerical integration equations (IEs) based on several quadrature rules (e.g. midpoint, trapezoidal, Simson's rule, and Gauss–Legendre [25]). The exact formula for defining the ARL require solving Fredholm IEs of the second kind [26]. For example, Crowder [27] developed an estimator for the ARL of a Gaussian process running on the standard EWMA control chart using the IE approach. Champ and Rigdon [28] defined the ARL of processes on the CUSUM and EWMA control charts using this method and compared the findings to those obtained using the Markov chain approach. The Fredholm IE of the second kind has been utilized to define the ARL for control charts; see [29,30,31,32,33]). Recently, Silpakob et al. [34] used the Fredholm IE of the second kind to derive the exact formula for the ARL of a process running on the modified EWMA control chart. The key aim of the present study is to derive the by using analytical IE as an exact formula for detecting a change in the mean of an ARFI(p, d) process with exponential white noise running on the modified EWMA control chart

The remainder of the article is outlined as follows: We provide a brief description of the ARFI(p, d) process with exponential white noise and the modified EWMA scheme in Section 2. Additionally, sensitivity analysis of the proposed design to detect changes in the mean of an ARFI(p, d) process with exponential white noise running on the modified EWMA control chart is presented. The analytical and approximated ARLs derived using the exact formula and the numerical IE technique for an ARFI(p, d) process with exponential white noise running on the modified EWMA control chart and the algorithm used to test them are provided in Section 3. Construction of the control limits, the results of the study, and a comparison of the proposed exact formula with existing ones are discussed in Section 4. Section 5 presents the real-life applicability of the proposed exact formula on the modified EWMA and existing charts. Concluding remarks and our future research direction are highlighted in Section 6. Finally, Appendix A, Appendix B and Appendix C show some important derivations used in the study.

2. Preliminaries

Here, we provide brief details about the ARFI(p, d) model with exponential white noise and the modified EWMA scheme.

2.1. The ARFI(p, d) Model

In the $\mathrm{ARFI}(p,d)$ process, parameters p and d are non-negative integers denoting the order of the AR process and the degree of fractional differencing, respectively [14,15]. It is a simple form of the $\mathrm{ARFIMA}(p,d,q)$ model, where $q=0$ and $d$ is fractional.

The long-memory $\mathrm{ARFI}(p,d)$ process is defined as

$$(1-{\varphi }_{1}B-{\varphi }_2{B}^2-\dots -{\varphi }_p{B}^p){(1-B)}^d{Z}_t\hspace{0.17em}\hspace{0.17em}=\vartheta +{\epsilon }_t,$$

(1)

where ${\varphi }_i$ is the i-th AR coefficient, $\vartheta$ is a constant term for the time series, and the exponential white noise process $\left\{{\epsilon }_t\right\}$ consists of independent identically distributed (i.i.d) random variables with mean $1/\nu$ and variance ${1/\nu }^2;\hspace{0.17em}\hspace{0.17em}{\epsilon }_t\sim Exp(\nu ).$ Since $d$ can be any real number, $B$ is referred to as a backward-shift operator; i.e.,$\hspace{0.17em}{B}^p{X}_t=X_{t-p}.$

For a stationary process, $d$ is in the range of 0 to 0.5; see for further details McLeod and Hipel [35]. Hence, ${(1-B)}^d$ can be expanded as a binomial series for the powers of the backward-shift operator as follows:

$${(1-B)}^d={\displaystyle \sum _{p=0}^{\infty }\left(\begin{array}{l}d\\ p\end{array}\right){(-B)}^p}=1-dB+\frac{d(d-1)B^2}{2!}-\frac{d(d-1)(d-2)B^3}{3!}+\dots ,\hspace{0.17em}$$

(2)

Equations $(1)$ and (2) can be rearranged for a generalized long-memory $\mathrm{ARFI}(p,d)$ process running on the modified EWMA control chart as follows:

$$\begin{array}{ll}{Z}_t=& \hspace{0.17em}\vartheta +{\epsilon }_t+\left(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\frac{d(d-1)(d-2)Z_t{}_{-3}}{3!}-\dots \right)\\ & +{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\frac{d(d-1)(d-2)}{3!}{\varphi }_i{Z}_{t-(i+3)}+\dots \right)},\end{array}$$

(3)

where ${\varphi }_i,i=1,2,\dots ,p$ are the coefficients for the AR component $(-1\le {\varphi }_i\le 1)$ and the initial values of $Z_0,Z_{t-1},Z_{t-2},Z_{t-3},\dots ,\hspace{0.17em}{Z}_{t-(p+1)}\hspace{0.17em}$ are 1.

2.2. The Modified EWMA Control Chart

Let $Z_t,\hspace{0.17em}\hspace{0.17em}t=1,\hspace{0.17em}2,\dots \hspace{0.17em}\hspace{0.17em},\hspace{0.17em}$ be the sequence of a generalized long-memory $\mathrm{ARFI}(p,d)$ process with exponential white noise. The modified EWMA statistic (Khan et al. [8]) for detecting small-to-moderate shifts in the process mean can be expressed as the following recursive equation:

$$Y_t=(1-\lambda )Y_{t-1}+\lambda Z_t+k(Z_t-Z_{t-1}),\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=1,\hspace{0.17em}2,\dots \hspace{0.17em}\hspace{0.17em},\hspace{0.17em}$$

(4)

where $k$ is an additional parameter, $Y_t$ represents the current value of the modified EWMA statistic, and $Y_{t-1}=Y_0$ denotes the previous value of the modified EWMA statistic given that $Y_0=u$ is the initial value and $Z_{t-1}=Z_0$ denotes the previous value of the process. The range of exponential smoothing parameter $\lambda$ is $0<\lambda \le 1,$. The modified EWMA statistic performs sensitively for small shifts in the process mean and small values of $\lambda$.

Remark 1.

The term$k(Z_t-Z_{t-1})$of the modified EWMA statistic is set with $k=0.$In addition, the modified EWMA control chart is the same as the Shewhart control chart when$\lambda =1.$

Meanwhile, ${\mu }_0$ and ${\sigma }^2$ are the target mean and variance of $Y_t$, respectively, and $Y_0={\mu }_0.$ Hence, the UCL and LCL of the modified EWMA control chart, respectively, become

$$\begin{array}{l}\mathrm{UCL}={\mu }_0+W\sigma \sqrt{\frac{(\lambda +2\lambda k+2k^2)}{2-\lambda }}\hspace{0.17em},\\ \mathrm{LCL}={\mu }_0-W\sigma \sqrt{\frac{(\lambda +2\lambda k+2k^2)}{2-\lambda }}\hspace{0.17em},\end{array}$$

(5)

where $W>0$ is a positive coefficient that determines the control limit width (the control limit coefficient), and ${\mu }_0$ and $\sigma$ represent the target mean and standard deviation of process, respectively, when the process is in control. The centerline (CL) of the modified EWMA control chart is the target value (usually the same as ${\mu }_0,$).

Remark 2.

The modified EWMA control chart is the same as the standard EWMA control chart when$k=0.$

2.3. The Design of the Modified EWMA Scheme

Here, the design of the modified EWMA scheme for running an $\mathrm{ARFI}(p,d)$ process with exponential white noise is covered.

Let ${\epsilon }_t,\hspace{0.17em}\hspace{0.17em}t=1,2,\dots ,$ be a sequence of continuous i.i.d. random variables from an exponential distribution with parameter $\nu .$ Now, the process is in control when $\nu ={\nu }_0,$ and out of control when $\nu ={\nu }_1.$ The change point models for ${\epsilon }_t$ are as follows:

$${\epsilon }_t=\left\{\begin{array}{l}Exp({\nu }_0),\hspace{0.17em}\hspace{0.17em}t=1,\hspace{0.17em}\hspace{0.17em}2,\hspace{0.17em}\hspace{0.17em}\dots ,\hspace{0.17em}\hspace{0.17em}m-1\hspace{0.17em}\hspace{0.17em}(no change)\\ Exp({\nu }_1>{\nu }_0),\hspace{0.17em}\hspace{0.17em}t=m,\hspace{0.17em}\hspace{0.17em}m+1,\hspace{0.17em}\hspace{0.17em}\dots (change),\end{array}\right.$$

(6)

where change point $m$ provides the sequence of ${\epsilon }_t$ for the in-control state ($t<m$) and the out-of-control state $(t\ge m).$ Following on from Equation (3), the $\mathrm{ARFI}(p,d)$ process $(Z_t)$ can be substituted into Equation (4). Thereby, the modified EWMA statistic in (4) is derived as

$$\begin{array}{ll}{Y}_t=& (1-\lambda )Y_{t-1}+\lambda Z_t+k(Z_t-Z_{t-1})\\ Y_t=& (1-\lambda )Y_{t-1}+\lambda \left(\vartheta \right.+{\epsilon }_t+d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots +{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\left.\dots )\right)}\\ & +k\left(\vartheta \right.+{\epsilon }_t+d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots +{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-.\left...)\right)}-k{Z}_{t-1},\hspace{0.17em}\end{array}$$

where $Y_{t-1}=u\hspace{0.17em}$ and $Z_{t-1}=Z_0.$ Accordingly, the proposed modified EWMA statistic becomes

$$\begin{array}{ll}{Y}_t=& (1-\lambda )u+\left(\vartheta \right.+{\epsilon }_t+d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \\ & +{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \left.)\right)}.(\lambda +k)-k{Z}_0.\end{array}$$

(7)

The corresponding stopping time $({\tau }_{a,b})$ is arrived at if $Y_t$ exceeds a given predetermined threshold for the first time; i.e.,

$${\tau }_{a,b}=inf\left\{t>0\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}{Y}_t>b\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}{Y}_t<a\hspace{0.17em}\hspace{0.17em}\right\},\hspace{0.17em}\hspace{0.17em}$$

(8)

where $a\hspace{0.17em}$ and $b$ denote the LCL and UCL, respectively. For the in-control process, the modified EWMA statistic can be restructured in the error term $({\epsilon }_t)$, where $a<{\epsilon }_t<b.$ Subsequently, $Y_t$ can be rearranged according to ${\epsilon }_t$ as follows:

$$\begin{gathered} \hspace{0.17em}\left(\frac{a-(1-\lambda )u+k{Z}_0}{(\lambda +k)}-\vartheta -\left(d{Z}_t{}_{-1}\right.-\frac{1}{2!}d(d-1)Z_t{}_{-2}+\dots \right.\hspace{0.17em}\hspace{0.17em}\left.+{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{1}{2!}d(d-1){\varphi }_i{Z}_{t-(i+2)}\left.-\dots )\right)}\right)\hspace{0.17em}\hspace{0.17em} \\ <{\epsilon }_t<\hspace{0.17em}\hspace{0.17em} \\ \hspace{0.17em}\left(\frac{b-(1-\lambda )u+k{Z}_0}{(\lambda +k)}-\vartheta -\left(d{Z}_t{}_{-1}\right.-\frac{1}{2!}d(d-1)Z_t{}_{-2}+\dots \right.\left.+{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{1}{2!}d(d-1){\varphi }_i{Z}_{t-(i+2)}-\left.\dots )\right)}\right) \end{gathered}$$

Hence, for $L<{\epsilon }_t<H$, the structure of the modified EWMA scheme can now be used for the analytical and approximated ARLs in the following section.

3. Derivation of Analytical and Approximate ARLs for Process with Exponential White Noise on a Modified EWMA Control Chart

In this section, analytical and approximated ARLs are derived as the solution of the IE for a ARFI(p, d) process with exponential white noise on the modified EWMA control chart to detect changes in the process mean. Additionally, this will be used to construct a numerical algorithm in the next part.

To assess the modified EWMA control chart, the stochastic properties of the related stopping time $({\tau }_{a,b})$ need to be determined. Considering the change point in (6), the ARL can be defined more rigorously with $E_m(.)$ as the expectation for fixed change point $m.$ Therefore,

$$\mathrm{ARL}=\left\{\begin{array}{l}{E}_{\infty }({\tau }_{a,b}),\hspace{0.17em}\hspace{0.17em}\nu ={\nu }_0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}in-control\hspace{0.17em}state (\mathrm{AR}{\mathrm{L}}_0)\\ E_1({\tau }_{a,b}),\hspace{0.17em}\hspace{0.17em}\nu \ne {\nu }_0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}out-of-control\hspace{0.17em}state\hspace{0.17em}\left(\mathrm{AR}{\mathrm{L}}_1\right).\end{array}\right.$$

Let $\mathcal{L}(u)$ denote the $ARL$ for an ARFI(p, d) process running on the modified EWMA control chart when the initial value is u defined as

$$\mathcal{L}(u)=E_{\infty }({\tau }_{a,b}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Y}_0=u.$$

Thus,

$$\mathcal{L}(u)=1+{\displaystyle \underset{L}{\overset{H}{\int }}\mathcal{L}\left((\lambda +k)y+Y_t\right)}.f(y)dy.$$

Based on the above equation, the integral equation (IE) derived from the Fredholm IE of the second kind can be written in the form:

$$\begin{array}{ll}\mathcal{L}(u)=& 1+{\displaystyle \underset{L}{\overset{H}{\int }}\mathcal{L}}\left((\lambda +k)y+(1-\lambda )u-k{Z}_0+\vartheta (\lambda +k)\right.+\left(d{Z}_t{}_{-1}\right.-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \\ & +{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}\right.-d{\varphi }_i{Z}_{t-(i+1)}}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \left.)\right)(\lambda +k)f(y)dy.\end{array}$$

By solving the IE, the integral variable can be adjusted by setting

$$\begin{array}{ll}x=& (\lambda +k)y+(1-\lambda )u-k{Z}_0+\vartheta (\lambda +k)+\left(d{Z}_t{}_{-1}\right.-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \\ & +{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}\right.-d{\varphi }_i{Z}_{t-(i+1)}}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \left.)\right)(\lambda +k).\end{array}$$

Thus, $\mathcal{L}(u)$ can be reworked as

$$\mathcal{L}(u)=1+\frac{1}{(\lambda +k)}{\displaystyle \underset{L}{\overset{H}{\int }}\mathcal{L}}(x).f\left(\begin{array}{l}\frac{x-(1-\lambda )u+k{Z}_0}{(\lambda +k)}-\vartheta -(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots )\\ \left.+{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots )}\right)\end{array}\right)dx,$$

(9)

where $f(x)=\frac{1}{\nu }.exp(\frac{-x}{\nu }).$ Therefore,

$$\begin{array}{ll}\mathcal{L}(u)=& 1+\frac{1}{\nu (\lambda +k)}\left[exp\left\{\frac{(1-\lambda )u}{\nu (\lambda +k)}\right\}.exp\left\{\frac{-k{Z}_{t-1}}{\nu (\lambda +k)}\right\}\right.\\ & \times exp\left.\left\{\frac{1}{\nu }\left(\vartheta +(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots )\right)+\frac{1}{\nu }\left({\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots )}\right)\right\}\right]\\ & \times {\displaystyle \underset{L}{\overset{H}{\int }}\mathcal{L}(x)}\hspace{0.17em}\hspace{0.17em}exp\left\{\frac{-x}{\nu (\lambda +k)}\right\}dx\end{array}$$

(10)

In the next step, the IE in Equation $(10)$ is used to calculate the analytical and approximate ARLs for detecting a change in the mean of an ARFI(p, d) process running on the modified EWMA control chart.

3.1. The Analytical ARL Using the Exact Formula

Here, we offer a new alternative to the ARL computation obtained by solving the IE for the ARL of an ARFI(p, d) process running on the modified EWMA control chart.

First, the IE in Equation (10) can be converted by setting new variables as

$$\begin{array}{ll}C& ={\displaystyle \underset{L}{\overset{H}{\int }}\mathcal{L}(x)}\hspace{0.17em}\hspace{0.17em}exp\left\{\frac{-x}{\nu (\lambda +k)}\right\}dx \mathrm{and} D(u)=exp\left\{\frac{(1-\lambda )u}{\nu (\lambda +k)}\right\}.exp\left\{\frac{-k{Z}_{t-1}}{\nu (\lambda +k)}\right\}\\ & \times exp\left\{\frac{1}{\nu }\left(\vartheta +\left(d{Z}_t{}_{-1}-\left.\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \right)\right.\right.\left.+\frac{1}{\nu }{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \right)}\right)\right\}.\end{array}$$

Subsequently, this equation can be rewritten as

$$\mathcal{L}(u)=1+\frac{D(u)}{\nu (\lambda +k)}.C$$

(11)

By substituting $\mathcal{L}(x)$ from Equation (8) into constant $C,$ we obtain

$$\begin{array}{ll}C& =-\hspace{0.17em}\nu (\lambda +k)\left[exp\left\{\frac{-H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-L}{\nu (\lambda +k)}\right\}\right]\\ & \times \left(1+\frac{1}{\lambda }exp\right.\left\{\begin{array}{l}\frac{1}{\nu (\lambda +k)}\left(-k{Z}_0+(\lambda +k)\left(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \right.\right.\\ \left.+{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \right)}\right)\end{array}\right\}\\ & {\left.\times exp\left\{\frac{\vartheta }{\nu }\right\}.\left[exp\left\{\frac{-\lambda H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-\lambda L}{\nu (\lambda +k)}\right\}\right]\right)}^{-1}.\end{array}$$

(12)

The details of the calculation are shown in Appendix A.

By using the solution for $C$ to solve the IE for $\mathcal{L}(u)$ in Equation (11), we attain the exact formula for the ARL of an ARFI(p, d) process running on the modified EWMA control chart as

$$\begin{array}{c}\mathcal{L}(u)=1+\frac{1}{\nu (\lambda +k)}\left[exp\left\{\frac{(1-\lambda )u}{\nu (\lambda +k)}\right\}.exp\left\{\frac{-k{Z}_0}{\nu (\lambda +k)}\right\}\right..\\ \times exp\left.\left\{\frac{1}{\nu }\left(\left.\vartheta +\left(d{Z}_t{}_{-1}-\left.\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \right)\right.\right)\right.+\frac{1}{\nu }\left({\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \right)}\right.\right\}\right]\\ \times \left(-\hspace{0.17em}\nu (\lambda +k)\left[exp\left\{\frac{-H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-L}{\nu (\lambda +k)}\right\}\right]\right)\\ \times \left(1+\frac{1}{\lambda }exp\right.\left\{\begin{array}{l}\frac{1}{\nu (\lambda +k)}\left(-k{Z}_0+(\lambda +k).\left(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots \right.\right.\\ \left.\left.+{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \right)}\right)\right)\end{array}\right\}.{\left.exp\left\{\frac{\vartheta }{\nu }\right\}.\left[exp\left\{\frac{-\lambda H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-\lambda L}{\nu (\lambda +k)}\right\}\right]\right)}^{-1}\end{array}$$

(13)

From Equation (13) given $Q=d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots +{\displaystyle \sum _{i=1}^p\left({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \right)},$ we can rearrange the equation as follows:

$$\begin{array}{ll}\mathcal{L}(u)=& 1+\frac{\left(exp\left\{\frac{(1-\lambda )u}{\nu (\lambda +k)}\right\}exp\left\{\frac{-k{Z}_0}{\nu (\lambda +k)}\right\}exp\left\{\frac{\vartheta +Q}{\nu }\right\}\right)}{\nu (\lambda +k)}\\ & \times \frac{-\nu (\lambda +k)\left(exp\left\{\frac{-H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-L}{\nu (\lambda +k)}\right\}\right)}{1+\frac{1}{\lambda }exp\left\{\frac{\left(-k{Z}_0+(\lambda +k)\right.\left.Q\right)}{\nu (\lambda +k)}\right\}exp\left\{\frac{\vartheta }{\nu }\right\}.\left(exp\left\{\frac{-\lambda H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-\lambda L}{\nu (\lambda +k)}\right\}\right)}.\end{array}$$

(14)

Finally, the exact formula for the $ARL$ of an ARFI(p, d) process running on the modified EWMA control chart is

$$\begin{array}{ll}\mathcal{L}(u)& =1\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\left(\lambda exp\left\{\frac{(1-\lambda )u}{\nu (\lambda +k)}\right\}.\left(exp\left\{\frac{-H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-L}{\nu (\lambda +k)}\right\}\right)\right)\\ & \times \left(\lambda .exp\left\{\frac{k{Z}_0}{\nu (\lambda +k)}\right\}exp\right.\left\{-\frac{1}{\nu }\left(\vartheta +d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots +{\displaystyle \sum _{i=1}^p\left(\begin{array}{l}{\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}\\ +\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \end{array}\right)}\right)\right\}\\ & {\left.+exp\left\{\frac{-\lambda H}{\nu (\lambda +k)}\right\}-exp\left\{\frac{-\lambda L}{\nu (\lambda +k)}\right\}\right)}^{-1}\end{array}$$

(15)

This is an easy one-step calculation scheme. The details of the calculation are shown in Appendix B.

In the previous equation, the in-control state occurs when exponential parameter $\nu ={\nu }_0.$ Conversely, the out-of-control state occurs when exponential parameter ${\nu }_1>{\nu }_0.$ Hence, ARL₁ can be expressed as

$$\begin{array}{ll}{\mathrm{ARL}}_1& =1\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\left(\lambda exp\left\{\frac{(1-\lambda )u}{{\nu }_1(\lambda +k)}\right\}.\left(exp\left\{\frac{-H}{{\nu }_1(\lambda +k)}\right\}-exp\left\{\frac{-L}{{\nu }_1(\lambda +k)}\right\}\right)\right)\\ & \times \left(\lambda .exp\left\{\frac{k{Z}_0}{{\nu }_1(\lambda +k)}\right\}exp\right.\left\{-\frac{1}{{\nu }_1}\left(\vartheta +d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots +{\displaystyle \sum _{i=1}^p\left(\begin{array}{l}{\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}\\ +\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots \end{array}\right)}\right)\right\}\\ & {\left.+exp\left\{\frac{-\lambda H}{{\nu }_1(\lambda +k)}\right\}-exp\left\{\frac{-\lambda L}{{\nu }_1\lambda +k)}\right\}\right)}^{-1}\end{array}$$

(16)

The exact ARL in Equation (15) was assessed by using Banach’s fixed-point theorem to verify its existence and uniqueness, the procedure for which is presented in Appendix C.

3.2. The Approximate ARL Based on the Numerical IE

The numerical IE technique was used to verify the accuracy of the analytical ARL derived by using the exact formula. The approximate ARL is derived based on the IE in Equation (9) by applying numerical quadrature rules that can be computed using many methods; see for instance (cf., [25,26]), as cited therein. In this study, we used a composite Simpson’s rule method. The IE in Equation (9) is solved by using a $2m+1$ linear equation system on interval [L, H], the length of which is 2m. The weight of each point is determined as follows:

$$w_j= \left\{\begin{array}{l}\frac{4}{3}\left(\frac{H-L}{2m}\right)\hspace{1em};j=1,3,\dots ,2m,\\ \frac{2}{3}\left(\frac{H-l}{2m}\right)\hspace{1em};j=2,4,\dots ,2m-1,\\ \frac{1}{3}\left(\frac{H-L}{2m}\right)\hspace{1em};Other cases.\end{array}\right.$$

Hence, the numerical IE technique (${\mathcal{L}}_N(u)$) for the ARL of a change in the mean of an ARFI(p, d) process running on the modified EWMA control chart can be written as

$${\mathcal{L}}_N(u)\approx 1+\frac{1}{\lambda +k}{\displaystyle \sum _{j=1}^{2m+1}{w}_j.}\mathcal{L}(x_j)f\left(\begin{array}{l}\frac{x_j-(1-\lambda )u+k{Z}_0}{(\lambda +k)}-\vartheta -(d{Z}_t{}_{-1}-\frac{d(d-1)}{2!}{Z}_t{}_{-2}+\dots )\\ \left.+{\displaystyle \sum _{i=1}^p({\varphi }_i{Z}_t{}_{-i}-d{\varphi }_i{Z}_{t-(i+1)}+\frac{d(d-1)}{2!}{\varphi }_i{Z}_{t-(i+2)}-\dots )}\right)\end{array}\right).$$

(17)

3.3. The Algorithm for Computing the Exact ARL

The Algorithm 1 was used to find the control chart coefficient and result for the ARL.

Algorithm 1. The exact formula of ARL for a long-memory ARFI (p, d) process on a modified EWMA control chart

The algorithmic description of the exact formula for a long-memory ARFI(p, d) process running on a modified EWMA control chart is as follows.

Input:

Set the values of the model coefficients:

$\vartheta =1;$ ${\varphi }_1\hspace{0.17em}=0.1,\hspace{0.17em}\hspace{0.17em}{\varphi }_2\hspace{0.17em}=0.2;\hspace{0.17em}\hspace{0.17em}d=0.15, 0.30, 0.45$

Set the values of the model coefficients: $Z_0,Z_{t-1},Z_{t-2},Z_{t-3},\dots ,\hspace{0.17em}{Z}_{t-(p+1)}\hspace{0.17em}=1$

Set the parameters for the modified EWMA control chart: $\lambda$ = 0.05, 0.10, 0.20; k = 0.0, 0.2, 0.5, 1.0, 2.0

Set the mean: ${\nu }_0=1$ for the in-control ARL0$({\epsilon }_t\sim Exp(\nu )).$

Set the mean shifts: ${\nu }_1=(1+\delta ){\nu }_0;$ $\delta =0.01,\hspace{0.17em}0.05,\hspace{0.17em}\hspace{0.17em}0.10,\hspace{0.17em}\hspace{0.17em}0.50,\hspace{0.17em}\hspace{0.17em}1.00,\hspace{0.17em}\hspace{0.17em}3.00$

Output:

In-control ARL

1: Solve the ARFI model defined as $Z_t$ for a generalized long-memory ARFI(p, d) process with exponential white noise.

2: Determine the values of the process coefficients $({\varphi }_1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\varphi }_2\hspace{0.17em})$ and the initial values of    $Z_{t-1},Z_{t-2},Z_{t-3},\dots ,\hspace{0.17em}{Z}_{t-(p+1)}$.

3: Specify $\hspace{0.17em}{\nu }_0$ for the in-control exponential white noise; i.e., $\hspace{0.17em}({\xi }_t\sim Exp(\nu )).$

4: Compute modified EWMA statistic $Y_t$ for the long-memory ARFI(p, d) process given in Equation (7).

5: Determine the values of the control chart coefficient for various combinations of $\lambda$ and $k.$

6: Calculate UCL (H) corresponding to the desired ARL0 by using Equation (15).

7: Execute Steps 2–6 while in control $(\delta =0),$ and then calculate the value of $H$ corresponding to each $(\lambda ,\hspace{0.17em}k)$ combination.

Out-of-control ARL

8: Compute ARL1 for shifts in the process mean $({\nu }_1)$ as given in Equation (16), where H is attained as per steps 6 and 7.

9: Record the computational time for the first out-of-control ARL1 signal.

10: Repeat steps 6–9 and calculate ARL1 corresponding to the specific shift size $(\delta ).$

4. Performance Evaluation and Comparison

4.1. Construction of the Control Limits

We calculated UCL (H) based on the exact formula for the in-control process, with process mean $({\nu }_0=1)$ according to Equation (15). The pre-specified value of ARL₀ was set as 370. The control limits under various combinations of $\lambda$ and $k$ can be adjusted to match the pre-chosen value of ARL₀. For various parameter combinations, we chose $\lambda =$ 0.05, 0.10, and 0.20 for the modified EWMA control chart. Although the value of $k$ can be selected independently of the value of $\lambda ,$ we chose $k=$ 0.0, 0.02, 0.5, 1.0, 2.0, $\vartheta =$ 1, ${\varphi }_1=$ 0.10, ${\varphi }_2\hspace{0.17em}=$ 0.20, $d=$ 0.15, 0.30, 0.45, and $Z_0,Z_{t-1},Z_{t-2},Z_{t-3},\dots ,\hspace{0.17em}{Z}_{t-(p+1)}=1.$ From the results in

Table 1. The UCL (H) values for the modified EWMA control chart for ARL₀ = 370.

λ	k	Long-Memory Models
		$\mathbf{ARFI}(1, \mathit{d}) {\mathit{\varphi }}_1$$= 0.1 \mathbf{and} \mathit{\eta } = 0.1$			$\mathbf{ARFI}(2, \mathit{d}) {\mathit{\varphi }}_1$$= 0.1, {\mathit{\varphi }}_2$$= 0.2 \mathbf{and} \mathit{\eta } = 0.1$
		d = 0.15	d = 0.30	d = 0.45	d = 0.15	d = 0.30	d = 0.45
0.05	0.0	1.847423905	1.514830700	1.288933760	1.580654300	1.355604580	1.196054520
	0.2	2.067005240	1.730856440	1.502560820	1.797381280	1.569938430	1.408698370
	0.5	2.396440000	2.054914850	1.822998020	2.122499230	1.891442100	1.727652800
	1.0	2.945884130	2.595259160	2.357225970	2.664635220	2.427470164	2.259379681
	2.0	4.047244550	3.677779121	3.427118068	3.750859956	3.501075556	3.324117363
0.10	0.0	1.935118820	1.573871550	1.331945360	1.644881290	1.403065300	1.233255050
	0.2	2.175098680	1.805922530	1.558817820	1.878471920	1.631449530	1.458043980
	0.5	2.536442520	2.155034800	1.899957280	2.229957090	1.974915150	1.795979350
	1.0	3.142676800	2.739911780	2.470956350	2.818971162	2.549959640	2.361411141
	2.0	4.372174930	3.922586520	3.623381301	4.010690557	3.711187907	3.501740791
0.>20	0.0	2.132823260	1.704069900	1.425806400	1.787056760	1.506908620	1.314198020
	0.2	2.421379540	1.972166200	1.681405890	2.058996510	1.766089380	1.564949260
	0.5	2.862318560	2.380051070	2.069247950	2.473067910	2.159663890	1.945045230
	1.0	3.621426280	3.076799520	2.728667740	3.181411710	2.829723170	2.590134860
	2.0	5.252565020	4.547212732	4.105681310	4.681259070	4.233149313	3.931832113

, the modified EWMA control chart with $\lambda =$ 0.05 and $k=$ 0.2 and the standard EWMA control chart with $\lambda =$ 0.05 and $k=$ 0 running an ARFI(1, 0.15) process yielded H = >1.>847423905 and >2.>067005240, respectively. Moreover, the modified EWMA control chart produced a higher value of H irrespective of the value of λ. Moreover, H remained higher for large values of λ and k, compared to lower values of the two parameters. For example, using $\lambda =$ 0.20 and $k=$ 2.0 for the ARFI(1, 0.15) process provided H = >5.>252565020 while $\lambda =$ 0.05 and $k=$ 0.2, yielded H = >2.>067005240.

4.2. Performance of the Control Charts

All of the computational routines were implemented using the Mathematica program. Since modified EWMA statistic $Y_t$ is sensitive to changes in the mean of an ARFI(p, d) process, ARL₁ was considered to be the best tool for measuring the control chart’s sensitivity. In the study, ${\nu }_1$ denoting changes in the process mean was defined as ${\nu }_1=(1+\delta ){\nu }_0,$ when $\delta$ was set as 0.01, 0.05, 0.10, 0.50, 1.00, and 3.00, and ${\nu }_0$ denotes the in-control process mean. For this task, the shifts in the process mean were set to be small (δ < 1.00) and moderate (1.00 ≤ δ < 3.00) [36].

To compare the performance in terms of ARL₁ obtained from using the proposed formula and the numerical IE technique, we set ARL₀ = 370 and then calculated ARL₁ using Equations (16) and (17), respectively. ARL₁ in Equation (17) determined the number of division points as $m=$ 500. The smaller the ARL₁ value, the better the statistical performance.

Furthermore, the performance of the proposed formula was compared with the numerical IE technique in terms of computational time (s) and percentage accuracy expressed as

$$\%\mathrm{Accuracy}=100-\left|\frac{\mathcal{L}(u)-{\mathcal{L}}_N(u)}{\mathcal{L}(u)}\right|\times 100\%,$$

(18)

where $\mathcal{L}(u)$ and ${\mathcal{L}}_N(u)$ denote the ARL values obtained by using the exact formula and numerical IE techniques, respectively. A value greater than 95% means that the proposed formula provided an ARL₁ value close to that for the numerical IE technique, which indicates excellent agreement between them.

The boldface values in

Table 2. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(1, d) processes running on the modified EWMA control chart with $\lambda$ = 0.05.

Coefficients of Model			q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.15	0.00	Exact	159.452	<0.001	48.227	<0.001	25.576	<0.001	5.516	<0.001	3.094	<0.001	1.633	<0.001
				NIE	159.452	3.64	48.227	3.59	25.576	3.47	5.517	3.72	3.094	3.45	1.633	3.56
				%Acc	100.00		100.00		100.00		99.98		100.00		100.00
			0.20	Exact	145.165	<0.001	42.237	<0.001	22.410	<0.001	5.064	<0.001	2.937	<0.001	1.611	<0.001
				NIE	145.165	3.53	42.237	3.77	22.410	3.64	5.064	3.50	2.937	3.53	1.612	3.50
				%Acc	100.00		100.00		100.00		100.00		100.00		99.94
			0.50	Exact	124.713	<0.001	34.504	<0.001	18.373	<0.001	4.469	<0.001	2.724	<0.001	1.581	<0.001
				NIE	124.713	3.47	34.504	3.47	18.373	3.56	4.469	3.55	2.724	3.45	1.581	3.61
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	94.321	<0.001	24.505	<0.001	13.219	<0.001	3.663	<0.001	2.419	<0.001	1.533	<0.001
				NIE	94.321	3.58	24.505	3.50	13.219	3.47	3.663	3.64	2.419	3.50	1.533	3.80
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	50.243	<0.001	12.391	<0.001	7.018	<0.001	2.572	<0.001	1.962	<0.001	1.448	<0.001
				NIE	50.243	3.63	12.391	3.64	7.018	3.55	2.572	3.45	1.962	3.63	1.448	3.59
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
		0.30	0.00	Exact	136.014	<0.001	38.555	<0.001	20.139	<0.001	4.635	<0.001	2.706	<0.001	1.520	<0.001
				NIE	136.014	3.52	38.555	3.44	20.139	3.48	4.635	3.67	2.706	3.78	1.520	3.59
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	122.063	<0.001	33.480	<0.001	17.777	<0.001	4.260	<0.001	2.576	<0.001	1.502	<0.001
				NIE	122.063	3.58	33.480	3.70	17.777	3.84	4.260	3.55	2.576	3.45	1.502	3.94
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	102.606	<0.001	27.019	<0.001	14.463	<0.001	3.768	<0.001	2.398	<0.001	1.477	<0.001
				NIE	102.606	3.45	27.019	3.49	14.463	3.48	3.768	3.47	2.398	3.61	1.477	3.45
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	74.838	<0.001	18.831	<0.001	10.292	<0.001	3.106	<0.001	2.144	<0.001	1.436	<0.001
				NIE	74.838	3.47	18.831	3.55	10.292	3.49	3.106	3.64	2.144	3.66	1.436	3.72
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	37.181	<0.001	9.226	<0.001	5.392	<0.001	2.218	<0.001	1.767	<0.001	1.366	<0.001
				NIE	37.181	3.44	9.226	3.61	5.392	3.52	2.219	3.64	1.767	3.56	1.366	3.63
				%Acc	100.00		100.00		100.00		99.95		100.00		100.00
		0.45	0.00	Exact	123.548	<0.001	33.849	<0.001	17.866	<0.001	4.148	<0.001	2.475	<0.001	1.447	<0.001
				NIE	123.548	3.47	33.849	3.45	17.866	3.59	4.148	3.56	2.475	3.52	1.447	3.66
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	110.073	<0.001	29.289	<0.001	15.538	<0.001	3.819	<0.001	2.361	<0.001	1.432	<0.001
				NIE	110.073	3.58	29.289	3.70	15.538	3.50	3.819	3.47	2.361	3.48	1.432	3.86
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	91.547	<0.001	23.521	<0.001	12.611	<0.001	3.388	<0.001	2.205	<0.001	1.409	<0.001
				NIE	91.547	3.48	23.522	3.53	12.611	3.45	3.388	3.47	2.205	3.48	1.409	3.51
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	65.666	<0.001	16.282	<0.001	8.950	<0.001	2.809	<0.001	1.984	<0.001	1.374	<0.001
				NIE	65.666	3.45	16.282	3.83	8.950	3.48	2.809	3.44	1.984	3.48	1.374	3.53
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	31.708	<0.001	7.914	<0.001	4.697	<0.001	2.037	<0.001	1.656	<0.001	1.313	<0.001
				NIE	31.708	3.69	7.914	3.56	4.697	3.44	2.037	3.69	1.656	3.64	1.313	3.55
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00

,

Table 3. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(1, d) processes running on the modified EWMA control chart with $\lambda$ = 0.10.

Coefficients of Model			q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.15	0.00	Exact	155.369	<0.001	46.464	<0.001	24.643	<0.001	5.385	<0.001	3.048	<0.001	1.626	<0.001
				NIE	155.369	3.45	46.464	3.67	24.643	3.63	5.385	3.48	3.048	3.61	1.626	3.56
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	144.283	<0.001	41.887	<0.001	22.229	<0.001	5.044	<0.001	2.934	<0.001	1.614	<0.001
				NIE	144.283	3.56	41.887	3.47	22.229	3.47	5.044	3.67	2.934	3.56	1.614	3.69
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	128.546	<0.001	35.880	<0.001	19.088	<0.001	4.586	<0.001	2.775	<0.001	1.596	<0.001
				NIE	128.546	3.47	35.880	3.52	19.088	3.50	4.586	3.53	2.775	3.44	1.596	3.56
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	105.241	<0.001	27.885	<0.001	14.941	<0.001	3.948	<0.001	2.543	<0.001	1.568	<0.001
				NIE	105.241	3.59	27.885	3.63	14.941	3.47	3.948	3.66	2.543	3.70	1.568	3.55
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	71.247	<0.001	17.719	<0.001	9.674	<0.001	3.038	<0.001	2.181	<0.001	1.518	<0.001
				NIE	71.247	3.53	17.719	3.61	9.674	3.63	3.038	3.44	2.181	3.72	1.518	3.72
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
		0.30	0.00	Exact	130.888	<0.001	36.635	<0.001	19.399	<0.001	4.485	<0.001	2.649	<0.001	1.509	<0.001
				NIE	130.888	3.48	36.635	3.52	19.399	3.49	4.485	3.53	2.649	3.49	1.509	3.64
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	119.429	<0.001	32.567	<0.001	17.307	<0.001	4.192	<0.001	2.551	<0.001	1.499	<0.001
				NIE	119.429	3.53	32.567	3.45	17.307	3.52	4.192	3.44	2.551	3.47	1.499	3.86
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	103.389	<0.001	27.276	<0.001	14.603	<0.001	3.799	<0.001	2.415	<0.001	1.483	<0.001
				NIE	103.389	3.44	27.278	3.56	14.603	3.56	3.799	3.55	2.415	3.53	1.483	3.73
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	80.092	<0.001	20.317	<0.001	11.061	<0.001	3.253	<0.001	2.214	<0.001	1.458	<0.001
				NIE	80.092	3.53	20.317	3.63	11.061	3.48	3.253	3.64	2.214	3.56	1.458	3.84
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	46.549	<0.001	11.496	<0.001	6.567	<0.001	2.473	<0.001	1.900	<0.001	1.413	<0.001
				NIE	46.549	3.70	11.496	3.55	6.567	3.59	2.473	3.58	1.900	3.56	1.413	3.84
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
		0.45	0.00	Exact	118.103	<0.001	31.950	<0.001	16.885	<0.001	3.998	<0.001	2.417	<0.001	1.435	<0.001
				NIE	118.103	3.56	31.950	3.61	16.885	3.61	3.998	3.45	2.417	3.52	1.435	3.48
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	106.848	<0.001	28.240	<0.001	15.002	<0.001	3.738	<0.001	2.330	<0.001	1.426	<0.001
				NIE	106.848	3.47	28.240	3.62	15.002	3.53	3.738	3.67	2.330	3.53	1.426	3.50
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	91.271	<0.001	23.447	<0.001	12.578	<0.001	3.389	<0.001	2.208	<0.001	1.412	<0.001
				NIE	91.271	3.56	23.447	3.67	12.578	3.48	3.389	3.45	2.208	3.67	1.412	3.70
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	69.046	<0.001	17.198	<0.001	9.425	<0.001	2.904	<0.001	2.031	<0.001	1.389	<0.001
				NIE	69.046	3.48	17.198	3.55	9.425	3.44	2.904	3.87	2.031	3.55	1.390	3.53
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	37.989	<0.001	9.404	<0.001	5.474	<0.001	2.216	<0.001	1.753	<0.001	1.349	<0.001
				NIE	37.989	3.61	9.404	3.56	5.474	3.47	2.216	3.44	1.753	3.45	1.349	3.64
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00

,

Table 4. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(1, d) processes running on the modified EWMA control chart with $\lambda$ = 0.20.

Coefficients of Model			q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.15	0.00	Exact	150.313	<0.001	44.356	<0.001	23.539	<0.001	5.238	<0.001	3.001	<0.001	1.622	<0.001
				NIE	150.313	3.77	44.356	3.67	23.539	3.56	5.238	3.44	3.001	3.44	1.622	3.67
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	145.627	<0.001	42.436	<0.001	22.527	<0.001	5.107	<0.001	2.968	<0.001	1.628	<0.001
				NIE	145.627	3.72	42.436	3.66	22.527	3.59	5.107	3.55	2.968	3.44	1.628	3.66
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	140.469	<0.001	40.340	<0.001	21.404	<0.001	4.950	<0.001	2.929	<0.001	1.639	<0.001
				NIE	140.469	3.45	40.340	3.81	21.404	3.58	4.950	3.42	2.929	3.53	1.639	3.53
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	139.377	<0.001	39.600	<0.001	20.858	<0.001	4.815	<0.001	2.898	<0.001	1.661	<0.001
				NIE	139.377	3.63	39.600	3.45	20.858	3.56	4.815	3.61	2.898	3.70	1.661	3.75
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	286.635	<0.001	100.019	<0.001	42.036	<0.001	5.463	<0.001	3.046	<0.001	1.732	<0.001
				NIE	286.618	3.55	100.018	3.56	42.036	3.42	5.463	3.42	3.046	3.48	1.732	3.55
				%Acc	99.99		100.00		100.00		100.00		100.00		100.00
		0.30	0.00	Exact	123.119	<0.001	33.837	<0.001	17.950	<0.001	4.270	<0.001	2.571	<0.001	1.495	<0.001
				NIE	123.119	3.66	33.837	3.69	17.950	3.47	4.270	3.55	2.571	3.52	1.495	3.58
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	116.176	<0.001	31.466	<0.001	16.749	<0.001	4.119	<0.001	2.529	<0.001	1.498	<0.001
				NIE	116.176	3.59	31.466	3.72	16.749	3.52	4.119	3.52	2.529	3.48	1.498	3.64
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	106.702	<0.001	28.367	<0.001	15.180	<0.001	3.915	<0.001	2.472	<0.001	1.503	<0.001
				NIE	106.702	3.67	28.367	3.48	15.180	3.59	3.915	3.45	2.472	3.80	1.503	3.55
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	93.710	<0.001	24.337	<0.001	13.139	<0.001	3.636	<0.001	2.393	<0.001	1.512	<0.001
				NIE	93.710	3.45	24.337	3.56	13.139	3.53	3.636	3.56	2.393	3.63	1.512	3.69
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			2.00	Exact	81.662	<0.001	20.667	<0.001	11.203	<0.001	3.322	<0.001	2.304	<0.001	1.542	<0.001
				NIE	81.662	3.52	20.667	3.88	11.203	3.59	3.322	3.48	2.304	3.61	1.542	3.83
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
		0.45	0.00	Exact	109.513	<0.001	29.072	<0.001	15.405	<0.001	3.772	<0.001	2.330	<0.001	1.418	<0.001
				NIE	109.513	3.61	29.072	3.55	15.405	3.47	3.772	3.52	2.330	3.48	1.418	3.63
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.20	Exact	102.125	<0.001	26.742	<0.001	14.239	<0.001	3.625	<0.001	2.288	<0.001	1.419	<0.001
				NIE	102.125	3.66	26.742	3.76	14.239	3.50	3.625	3.47	2.288	3.52	1.419	3.55
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.50	Exact	91.894	<0.001	23.653	<0.001	12.695	<0.001	3.423	<0.001	2.230	<0.001	1.421	<0.001
				NIE	91.894	3.44	23.653	3.48	12.695	3.58	3.423	3.58	2.230	3.52	1.421	3.53
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			1.00	Exact	77.186	<0.001	19.463	<0.001	10.601	<0.001	3.136	<0.001	2.144	<0.001	1.425	<0.001
				NIE	77.186	3.41	19.464	3.53	10.601	3.48	3.136	3.48	2.144	3.64	1.425	3.53
				%Acc	100.00		99.99		100.00		100.00		100.00		100.00
			2.00	Exact	56.629	<0.001	14.028	<0.001	7.867	<0.001	2.725	<0.001	2.015	<0.001	1.439	<0.001
				NIE	56.629	3.55	14.028	3.62	7.867	3.42	2.725	3.84	2.015	3.66	1.439	3.59
				%Acc	100.00		100.00		100.00		100.00		100.00		100.00

,

Table 5. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(2, d) processes running on the modified EWMA control chart with $\lambda$ = 0.05.

Coefficients of Model				q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.20	0.15	0.00	Exact	140.082	<0.001	40.153	<0.001	21.258	<0.001	4.791	<0.001	2.778	<0.001	1.542	<0.001
					NIE	140.082	4.37	40.153	4.51	21.258	4.75	4.791	4.59	2.778	4.53	1.542	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	126.021	<0.001	34.914	<0.001	18.539	<0.001	4.402	<0.001	2.642	<0.001	1.523	<0.001
					NIE	126.021	4.59	34.914	4.53	18.539	4.62	4.402	4.70	2.642	4.56	1.523	4.70
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	106.318	<0.001	28.228	<0.001	15.099	<0.001	3.891	<0.001	2.458	<0.001	1.497	<0.001
					NIE	106.318	4.62	28.228	4.50	15.100	4.65	3.891	4.45	2.458	4.45	1.497	4.69
					%Acc	100.00		100.00		99.99		100.00		100.00		100.00
				1.00	Exact	78.001	<0.001	19.727	<0.001	10.759	<0.001	3.203	<0.001	2.194	<0.001	1.455	<0.001
					NIE	78.001	4.75	19.727	4.58	10.760	4.55	3.203	4.48	2.194	4.61	1.455	4.62
					%Acc	100.00		100.00		99.99		100.00		100.00		100.00
				2.00	Exact	39.163	<0.001	9.703	<0.001	5.642	<0.001	2.279	<0.001	1.802	<0.001	1.382	<0.001
					NIE	39.163	4.45	9.703	4.55	5.642	4.69	2.279	4.50	1.802	4.53	1.382	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.30	0.00	Exact	127.019	<0.001	35.132	<0.001	18.559	<0.001	4.285	<0.001	2.541	<0.001	1.468	<0.001
					NIE	127.019	4.55	35.132	4.45	18.559	4.52	4.285	4.52	2.541	4.59	1.468	4.76
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	113.392	<0.001	30.427	<0.001	16.148	<0.001	3.943	<0.001	2.422	<0.001	1.452	<0.001
					NIE	113.392	4.48	30.427	4.61	16.148	4.44	3.943	4.44	2.422	4.52	1.452	4.73
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	94.580	<0.001	24.466	<0.001	13.113	<0.001	3.494	<0.001	2.260	<0.001	1.429	<0.001
					NIE	94.580	4.42	24.466	4.50	13.113	4.45	3.494	4.52	2.260	4.47	1.429	4.64
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	68.145	<0.001	16.964	<0.001	9.311	<0.001	2.892	<0.001	2.030	<0.001	1.392	<0.001
					NIE	68.145	4.42	16.964	4.67	9.311	4.70	2.892	4.55	2.030	4.53	1.392	4.55
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	33.149	<0.001	8.259	<0.001	4.881	<0.001	2.087	<0.001	1.687	<0.001	1.328	<0.001
					NIE	33.149	4.72	8.259	4.39	4.881	4.58	2.087	4.64	1.687	4.62	1.328	4.55
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.45	0.00	Exact	118.941	<0.001	32.178	<0.001	16.963	<0.001	3.964	<0.001	2.385	<0.001	1.417	<0.001
					NIE	118.941	4.66	32.178	4.53	16.963	4.50	3.964	4.56	2.385	4.58	1.417	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	105.693	<0.001	27.811	<0.001	14.745	<0.001	3.653	<0.001	2.277	<0.001	1.403	<0.001
					NIE	105.693	4.67	27.811	4.48	14.745	4.47	3.653	4.59	2.277	4.50	1.403	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	87.576	<0.001	22.301	<0.001	11.960	<0.001	3.246	<0.001	2.130	<0.001	1.382	<0.001
					NIE	87.576	4.50	22.301	4.56	11.960	4.55	3.246	4.48	2.130	4.52	1.382	4.73
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	62.460	<0.001	15.407	<0.001	8.485	<0.001	2.699	<0.001	1.922	<0.001	1.349	<0.001
					NIE	62.460	4.42	15.407	4.44	8.485	4.41	2.699	4.44	1.922	4.50	1.349	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	29.888	<0.001	7.478	<0.001	4.462	<0.001	1.971	<0.001	1.613	<0.001	1.292	<0.001
					NIE	29.888	4.47	7.478	4.56	4.462	4.42	1.971	4.45	1.613	4.44	1.292	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00

,

Table 6. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(2, d) processes running on the modified EWMA control chart with $\lambda$ = 0.10.

Coefficients of model				q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.20	0.15	0.00	Exact	135.095	<0.001	38.239	<0.001	20.257	<0.001	4.642	<0.001	2.723	<0.001	1.532	<0.001
					NIE	135.095	4.81	38.239	4.89	20.257	4.91	4.642	4.55	2.723	4.62	1.532	4.78
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	123.629	<0.001	34.066	<0.001	18.102	<0.001	4.340	<0.001	2.621	<0.001	1.521	<0.001
					NIE	123.629	4.50	34.066	4.41	18.102	4.50	4.340	4.58	2.621	4.45	1.521	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	107.523	<0.001	28.628	<0.001	15.312	<0.001	3.935	<0.001	2.480	<0.001	1.505	<0.001
					NIE	107.523	4.47	28.628	4.66	15.312	4.53	3.935	4.53	2.480	4.58	1.505	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	84.014	<0.001	21.453	<0.001	11.651	<0.001	3.370	<0.001	2.273	<0.001	1.479	<0.001
					NIE	84.014	4.56	21.453	4.47	11.651	4.66	3.370	4.62	2.273	4.62	1.479	4.62
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	49.881	<0.001	12.319	<0.001	6.989	<0.001	2.563	<0.001	1.949	<0.001	1.433	<0.001
					NIE	49.881	4.77	12.319	4.50	6.989	4.47	2.563	4.47	1.949	4.58	1.433	4.76
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.30	0.00	Exact	121.649	<0.001	33.222	<0.001	17.568	<0.001	4.134	<0.001	2.483	<0.001	1.457	<0.001
					NIE	121.649	4.65	33.222	4.67	17.568	4.47	4.134	4.59	2.483	4.48	1.457	4.77
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	110.311	<0.001	29.407	<0.001	15.625	<0.001	3.864	<0.001	2.393	<0.001	1.447	<0.001
					NIE	110.311	4.53	29.407	4.92	15.625	4.72	3.864	4.55	2.393	4.48	1.447	4.62
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	94.565	<0.001	24.470	<0.001	13.121	<0.001	3.503	<0.001	2.267	<0.001	1.432	<0.001
					NIE	94.565	4.47	24.470	4.58	13.121	4.64	3.503	4.67	2.267	4.67	1.432	4.61
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	71.985	<0.001	18.017	<0.001	9.857	<0.001	3.000	<0.001	2.083	<0.001	1.409	<0.001
					NIE	71.985	4.55	18.017	4.48	9.857	4.50	3.000	4.48	2.083	4.61	1.409	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	40.155	<0.001	9.931	<0.001	5.752	<0.001	2.285	<0.001	1.794	<0.001	1.368	<0.001
					NIE	40.155	4.66	9.931	4.51	5.752	4.55	2.285	4.69	1.794	4.48	1.368	4.73
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.45	0.00	Exact	113.414	<0.001	30.300	<0.001	15.996	<0.001	3.816	<0.001	2.327	<0.001	1.405	<0.001
					NIE	113.414	4.62	30.300	4.45	15.996	4.37	3.816	4.47	2.327	4.45	1.405	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	102.301	<0.001	26.734	<0.001	14.195	<0.001	3.569	<0.001	2.244	<0.001	1.396	<0.001
					NIE	102.301	4.44	26.734	4.69	14.195	4.44	3.569	4.47	2.244	4.44	1.396	4.39
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	86.990	<0.001	22.136	<0.001	11.880	<0.001	3.238	<0.001	2.130	<0.001	1.383	<0.001
					NIE	86.990	4.48	22.136	4.50	11.880	4.42	3.238	4.41	2.130	4.44	1.383	4.52
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	65.297	<0.001	16.165	<0.001	8.878	<0.001	2.779	<0.001	1.962	<0.001	1.362	<0.001
					NIE	65.297	4.44	16.165	4.44	8.878	4.47	2.779	4.55	1.962	4.49	1.362	4.52
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	35.335	<0.001	8.761	<0.001	5.133	<0.001	2.128	<0.001	1.699	<0.001	1.324	<0.001
					NIE	35.335	4.47	8.761	4.41	5.133	4.48	2.128	4.62	1.699	4.83	1.324	4.66
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00

and

Table 7. The ARL₁ results using the exact formula and the numerical IE technique for ARFI(2, d) processes running on the modified EWMA control chart with $\lambda$ = 0.20.

Coefficients of model				q	δ	0.01		0.05		0.10		0.50		1.00		3.00
$\mathit{\vartheta }$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d			ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time	ARL1	Time
1.00	0.10	0.20	0.15	0.00	Exact	127.681	<0.001	35.503	<0.001	18.838	<0.001	4.435	<0.001	2.647	<0.001	1.519	<0.001
					NIE	127.681	4.58	35.503	4.73	18.838	4.52	4.435	4.55	2.647	4.70	1.519	4.62
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	120.986	<0.001	33.154	<0.001	17.642	<0.001	4.284	<0.001	2.606	<0.001	1.522	<0.001
					NIE	120.986	4.70	33.154	4.55	17.642	4.58	4.284	4.66	2.606	4.91	1.522	4.59
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	111.957	<0.001	30.112	<0.001	16.094	<0.001	4.083	<0.001	2.551	<0.001	1.528	<0.001
					NIE	111.957	4.59	30.112	5.03	16.094	4.51	4.083	4.67	2.551	4.95	1.528	4.53
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	100.036	<0.001	26.286	<0.001	14.142	<0.001	3.816	<0.001	2.477	<0.001	1.539	<0.001
					NIE	100.036	4.61	26.286	4.72	14.142	4.52	3.816	4.66	2.477	4.51	1.539	4.81
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	94.358	<0.001	24.213	<0.001	12.923	<0.001	3.570	<0.001	2.411	<0.001	1.575	<0.001
					NIE	94.357	4.53	24.213	4.53	12.923	4.42	3.570	4.62	2.411	4.56	1.575	4.62
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.30	0.00	Exact	113.249	<0.001	30.351	<0.001	16.089	<0.001	3.910	<0.001	2.398	<0.001	1.440	<0.001
					NIE	113.250	4.73	30.351	4.47	16.089	4.66	3.910	4.47	2.398	4.50	1.440	4.73
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	105.939	<0.001	27.996	<0.001	14.907	<0.001	3.760	<0.001	2.356	<0.001	1.442	<0.001
					NIE	105.939	4.50	27.996	4.48	14.907	4.64	3.760	4.67	2.356	4.50	1.442	4.52
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	95.836	<0.001	24.879	<0.001	13.344	<0.001	3.557	<0.001	2.297	<0.001	1.444	<0.001
					NIE	95.836	4.53	24.879	4.67	13.344	4.62	3.557	4.51	2.297	4.56	1.444	4.73
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	81.408	<0.001	20.681	<0.001	11.239	<0.001	3.269	<0.001	2.212	<0.001	1.450	<0.001
					NIE	81.408	4.56	20.681	4.41	11.239	4.58	3.269	4.50	2.212	4.48	1.450	4.58
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	62.135	<0.001	15.450	<0.001	8.595	<0.001	2.871	<0.001	2.090	<0.001	1.468	<0.001
					NIE	62.135	4.55	15.450	4.73	8.595	4.52	2.871	4.59	2.090	4.59	1.468	4.56
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
			0.45	0.00	Exact	104.612	<0.001	27.427	<0.001	14.524	<0.001	3.590	<0.001	2.239	<0.001	1.387	<0.001
					NIE	104.612	4.56	27.427	4.52	14.524	4.66	3.590	4.44	2.239	4.47	1.387	4.50
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.20	Exact	97.172	<0.001	25.145	<0.001	13.388	<0.001	3.447	<0.001	2.198	<0.001	1.388	<0.001
					NIE	97.172	4.52	25.145	4.45	13.388	4.47	3.447	4.44	2.198	4.51	1.388	4.50
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				0.50	Exact	86.859	<0.001	22.115	<0.001	11.879	<0.001	3.250	<0.001	2.141	<0.001	1.389	<0.001
					NIE	86.859	4.50	22.115	4.45	11.879	4.47	3.250	4.44	2.141	4.45	1.389	4.59
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				1.00	Exact	71.970	<0.001	17.985	<0.001	9.823	<0.001	2.968	<0.001	2.055	<0.001	1.392	<0.001
					NIE	71.970	4.41	17.985	5.37	9.823	4.50	2.968	4.41	2.055	4.47	1.392	4.67
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00
				2.00	Exact	50.490	<0.001	12.465	<0.001	7.057	<0.001	2.550	<0.001	1.921	<0.001	1.402	<0.001
					NIE	50.490	4.45	12.465	4.44	7.057	4.34	2.550	4.45	1.921	4.42	1.402	4.53
					%Acc	100.00		100.00		100.00		100.00		100.00		100.00

indicate the smallest ARL₁ values obtained using the exact formula under various shifts in the mean of the process running on the modified EWMA control chart with various values of $\lambda$ when $k=$ 2.0, except for the ARFI(1, 0.15) process where we used $k=$ 2.0 and $k=$ 1.0 under various mean shifts. The main findings from Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 are as follows:

• The ARL₁ values tended to decline rapidly and monotonically as $\delta$ was increased.
• The ARL₁ values created by the exact formula were comparable to those approximated by using the numerical IE technique. The percentage accuracy results were around 99% in all cases, meaning that they are in good agreement.
• The computational time to calculate ARL₁ using the exact formula (less than 0.001 s) was substantially less than that required by the numerical IE technique (3–4 s) in all cases.
• For all settings of $\lambda$ and $k$, the ARL value of the ARFI(1, 0.45) process was shorter than those of the ARFI(1, 0.30) and ARFI(1, 0.15) processes (Figure 1a). The ARFI(2, d) process also yielded similar results to ARFI(1, d), with the ARL of ARFI(2, 0.45) being the shortest (Figure 1b).

To detect small-to-moderate shifts in the process mean, the modified EWMA control chart with small $\lambda$ and large $k$ performed better than with large $\lambda$ and large $k.$ Moreover, the modified EWMA control chart performed better than the standard EWMA control chart with various parameters of $\lambda$ when $k=0$ for all of the processes tested. Thus, the parameter combination of $\lambda =$ 0.05 and $k=$ 2.0 is recommended for practical applications.

In summary, the results show that the exact formula provides a good method for detecting shifts in the mean of an $\mathrm{ARFI}(p,d)$ process with exponential white noise running on the modified EWMA control chart. In particular, calculating the ARL is simple and the computational time is significantly reduced.

4.3. Evaluation of the Modified EWMA Control Chart

We compared the performances of the modified EWMA control chart and the standard EWMA control chart. The ARL of the modified control chart was determined using the exact formula. In addition, some other properties of the RL, such as the standard deviation of the run length (SDRL) and the median run length (RML), were also considered [37]. For the modified EWMA control chart, the in-control SDRL and MRL are, respectively, calculated by using

$${\mathrm{ARL}}_0=\frac{\mathrm{l}}{\alpha },\hspace{0.17em}\hspace{0.17em}{\mathrm{SDRL}}_0=\sqrt{\frac{1-\alpha }{{\alpha }^2}},\hspace{0.17em}\hspace{0.17em}{\mathrm{MRL}}_0=\frac{\log (0.5)}{\log (1-\alpha )},$$

(19)

where type 1 error (false alarm) $\alpha =1-P(L<Y_t<H\left|\hspace{0.17em}\nu \right.).$ The associated probability of a false alarm is referred to as the false alarm rate (FAR). In our study, a FAR value of 0.0027 provided ARL₀$\approx$370, SDRL₀$\approx$370, AND MRL₀$\approx$. In contrast, the out-of-control ARL, SDRL, and MRL values are, respectively, calculated by using

$${\mathrm{ARL}}_1=\frac{1}{(1-\beta )},\hspace{0.17em}\hspace{0.17em}{\mathrm{SDRL}}_1=\sqrt{\frac{\beta }{{(1-\beta )}^2}},\hspace{0.17em}\hspace{0.17em}{\mathrm{MRL}}_1=\frac{\log (0.5)}{\log \beta },$$

(20)

where the probability of type II error $\beta =P(L<Y_t<H{\left|\hspace{0.17em}\nu \right.}_1).$ ARL₁$\approx$1 is desired for large-size shifts in the process mean. The probabilities of the two types of error are the inverse and vice versa, which is consistent with the technique using the previously proposed formula (cf. Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7).

Here, we discuss the performance of the modified EWMA $(k=2)$ control chart compared to the standard EWMA $(k=0)$ control chart. The modified EWMA control chart attained smaller ARLs than the standard EWMA control chart. We calculated the ARL values obtained using the exact formula (see Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7), $\mathrm{SDRL}$, and $\mathrm{MRL}$ (see Equations (19) and (20)) for both control charts running the processes given in

Table 8. The comparative performances of the standard and modified EWMA control charts for various shift sizes when $\lambda$ = 0.05.

$\mathit{\delta }$		Long-Memory ARFI(1, d) Model						Long-Memory ARFI(2, d) Model
	d	0.15		0.30		0.45		0.15		0.30		0.45
	Control Chart	EWMA (k = 0)	MEWMA (k = 2)	EWMA (k = 0)	MEWMA (k = 2)	EWMA (k = 0)	MEWMA (k = 2)	EWMA (k = 0)	MEWMA (k = 2)	EWMA (k = 0)	MEWMA (k = 2)	EWMA (k = 0)	MEWMA (k = 2)
0.01	ARL1	159.4520	50.2430	136.0140	37.1810	123.5480	31.7080	140.0820	39.1630	127.0190	33.1490	118.9410	29.8880
	SDRL1	158.9512	49.7405	135.5131	36.6776	123.0470	31.2040	139.5811	38.6598	126.5180	32.6452	118.4399	29.3837
	MRL1	110.1768	34.4781	93.9307	25.4238	85.2899	21.6299	96.7505	26.7977	87.6958	22.6288	82.0966	20.3682
0.05	ARL1	48.2270	12.3910	38.5550	9.2260	33.8490	7.9140	40.1530	9.7030	35.1320	8.2590	32.1780	7.4780
	SDRL1	47.7244	11.8805	38.0517	8.7117	33.3453	7.3971	39.6498	9.1894	34.6284	7.7429	31.6741	6.9601
	MRL1	33.0806	8.2374	26.3762	6.0418	23.1140	5.1312	27.4839	6.3728	24.0034	5.3707	21.9557	4.8285
0.10	ARL1	25.5760	7.0180	20.1390	5.3920	17.8660	4.6970	21.2580	5.6420	18.5590	4.8810	16.9630	4.4620
	SDRL1	25.0710	6.4988	19.6326	4.8664	17.3588	4.1671	20.7520	5.1176	18.0521	4.3524	16.4554	3.9303
	MRL1	17.3791	4.5091	13.6098	3.3790	12.0339	2.8953	14.3856	3.5529	12.5143	3.0234	11.4078	2.7316
0.50	ARL1	5.5160	2.5720	4.6350	2.2180	4.1480	2.0370	4.7910	2.2790	4.2850	2.0870	3.9640	1.9710
	SDRL1	4.9910	2.0108	4.1047	1.6436	3.6136	1.4534	4.2618	1.7073	3.7518	1.5062	3.4277	1.3834
	MRL1	3.4653	1.4079	2.8521	1.1564	2.5127	1.0267	2.9608	1.1999	2.6082	1.0626	2.3843	0.9791
1.00	ARL1	3.0940	1.9620	2.7060	1.7670	2.4750	1.6560	2.7780	1.8020	2.5410	1.6870	2.3850	1.6130
	SDRL1	2.5454	1.3738	2.1486	1.1642	1.9107	1.0423	2.2224	1.2022	1.9788	1.0766	1.8175	0.9944
	MRL1	1.7755	0.9726	1.5025	0.8306	1.3392	0.7485	1.5533	0.8562	1.3859	0.7716	1.2753	0.7164
3.00	ARL1	1.6330	1.4480	1.5200	1.3660	1.4470	1.3130	1.5420	1.3820	1.4680	1.3280	1.4170	1.2920
	SDRL1	1.0167	0.8054	0.8890	0.7071	0.8042	0.6411	0.9142	0.7266	0.8289	0.6600	0.7687	0.6142
	MRL1	0.7314	0.5908	0.6462	0.5263	0.5901	0.4834	0.6629	0.5391	0.6063	0.4957	0.5667	0.4661

. The control chart performance with the lowest $\mathrm{ARL}, \mathrm{SDRL} \mathrm{and} \mathrm{MRL}$ provided the best performance. We conclude that the $\mathrm{MRL}$ and $\mathrm{SDRL}$ values were smaller than the $\mathrm{ARL}$ values in all cases. Overall, based on the results in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7, $\lambda =0.05\hspace{0.17em}$ is recommended.

For $\lambda =$ 0.05, we can see that the value of ${\mathrm{ARL}}_1$ on the modified EWMA control chart was smaller than on the standard EWMA control chart for every value of the shift parameter. Thus, the modified EWMA control chart is more sensitive than the EWMA control chart in detecting small-to-moderate shifts in the mean of long-memory processes.

We noticed that the modified EWMA control chart running the $\mathrm{ARFI}(1,\hspace{0.17em}\hspace{0.17em}d=0.45)$ process did have good detection ability for small-to-moderate shifts in the process mean and performed better than with the $\mathrm{ARFI}(1,\hspace{0.17em}\hspace{0.17em}d=0.30)$ and $\mathrm{ARFI}(1,\hspace{0.17em}\hspace{0.17em}d=0.15)$ processes. In addition, as the value of $d$ increased for the $\mathrm{ARFI}(2,\hspace{0.17em}\hspace{0.17em}d)$ processes, so did the ARL₁ values of the modified EWMA control chart performed when $d$ was small. Finally, for $k=$2 and $\lambda =$ 0.05, the detection ability of the modified EWMA control chart for small-to-moderate shifts in the mean of $\mathrm{ARFI}(2,\hspace{0.17em}\hspace{0.17em}d=0.45)$ processes were very sensitive. Thus, it can be concluded that a small smoothing parameter value is appropriate for detecting smaller-to-moderate shifts in the mean of an $\mathrm{ARFI}(p,d)$ process running on the modified EWMA control chart using the exact formula.

5. Application of the Proposed Exact Formula for the ARL

Here, we illustrate the applicability of the proposed formula on two real-life datasets following $\mathrm{ARFI}(p,\hspace{0.17em}\hspace{0.17em}d)$ processes with exponential white noise running on the modified EWMA control chart. The first dataset comprises the incidence of pneumonia in Thailand and the second one comprises the Netflix Inc. stock price. Since time-series modeling decomposes the actual series into fitted values and residuals, we used the Eviews 10 statistical software package to fit and evaluate the parameters of the $\mathrm{ARFI}(1,\hspace{0.17em}\hspace{0.17em}d)$ and $\mathrm{ARFI}(2,\hspace{0.17em}\hspace{0.17em}d)$ models. Thereafter, whether the residuals behaving as white noise were exponentially distributed was tested using the Kolmogorov–Smirnov test in the SPSS software package.

Application 1.

The monthly incidence of pneumonia per 100,000 people from January 2014 until September 2022 was obtained from the Ministry of Public Health in Thailand. The dataset consisting of 105 observations was downloaded from https://hdcservice.moph.go.th accessed on 1 October 2022. The following$ARFI(1,\hspace{0.17em}\hspace{0.17em}0.49999)$model was found to be suitable:

$$\begin{array}{ll}{Z}_t=& {\epsilon }_t+0.49999Z_t{}_{-1}+0.1250Z_{t-2}+0.0625Z_{t-}{}_3+{\varphi }_1{Z}_{t-}{}_1-\hspace{0.17em}0.49999{\varphi }_2{Z}_{t-}{}_2\\ & -0.1250{\varphi }_1{Z}_{t-}{}_3-0.0625{\varphi }_1{Z}_{t-}{}_4,\end{array}$$

(21)

where${\epsilon }_t\sim Exp({\nu }_0=23,379.0812),$and AR coefficient  ${\varphi }_1=0.26997.$

Application 2.

The daily Netflix Inc. (NFLX: NASDAQ) stock price from 17 February 2015 to 25 January 2018 were downloaded from https://www.investing.com/equities/netflix,-inc.-historical-data accessed on 1 October 2022. The dataset consisting of 743 observations was fitted to an$ARFI(2,\hspace{0.17em}\hspace{0.17em}0.074535)$model as follows:

$$\begin{array}{ll}{Z}_t=& {\epsilon }_t+0.07454Z_t{}_{-1}+0.03449Z_{t-}{}_2+0.02214Z_{t-}{}_3+{\varphi }_1{Z}_{t-}{}_1-\hspace{0.17em}0.07454{\varphi }_1{Z}_{t-}{}_2-\hspace{0.17em}\hspace{0.17em}0.03449{\varphi }_1{Z}_{t-}{}_3\\ & -\hspace{0.17em}0.02214{\varphi }_1{Z}_{t-}{}_4+{\varphi }_2{Z}_{t-}{}_2-0.07454{\varphi }_2{Z}_{t-}{}_3-0.03449{\varphi }_2{Z}_{t-}{}_4-0.02214{\varphi }_2{Z}_{t-}{}_5-\hspace{0.17em}0.07454{\varphi }_1{Z}_{t-}{}_2\\ & +{\varphi }_2{Z}_{t-}{}_2+0.07454Z_t{}_{-1}+{\varphi }_1{Z}_{t-}{}_1,\end{array}$$

where${\epsilon }_t\sim Exp({\nu }_0=2.30598),\hspace{0.17em}\hspace{0.17em}{\varphi }_1=1.009622\hspace{0.17em}$and${\varphi }_2=-0.009622.$

In both applications, we used Equation (15) in conjunction with Equations (21) and (22) to compute the UCLs for ARL₀ = 370. Initially, we focused on the one-sided upper modified EWMA control chart with an upper control limit (UCL) for detecting shifts in the mean of a process. Previously, researchers have suggested $\lambda =0.05\hspace{0.17em}$ with $k=2.0$ as appropriate for the modified EWMA control chart. These yield $\mathrm{UCL}\hspace{0.17em}\left(H\right)$ calculations of 27,878.4119 and 3.00079 for the $\mathrm{ARFI}(1,\hspace{0.17em}\hspace{0.17em}0.49999)$ and $\mathrm{ARFI}(2,\hspace{0.17em}\hspace{0.17em}0.074535)$ models, respectively. Moreover, CL = 23,379.0812, LCL = 0 and CL = 2.30598, LCL = 0, respectively, as shown in Figure 1. The ARL performances obtained from using the exact formula and numerical IE technique were compared against SDRL and MRL benchmarks (

Table 9. Comparative ARL, SDRL, and MRL performance of EWMA and Modified EWMA charts for various shifts when $\lambda$ = 0.05.

δ	Applications 1				Applications 2
	ARL1		SDRL1	MRL1	ARL1		SDRL1	MRL1
	Exact	NIE			Exact	NIE
0.01	60.4034	60.4034	59.9013	41.5209	28.556	28.5556	28.0515	19.4449
	(<0.001)	(9.063)			(<0.001)	(11.015)
0.05	14.8503	14.8503	14.3416	9.9428	6.9413	6.9413	6.4219	4.4558
	(<0.001)	(8.640)			(<0.001)	(11.485)
0.10	8.1881	8.18812	14.3416	5.3215	4.0418	4.0418	4.0418	2.4386
	(<0.001)	(8.891)			(<0.001)	(11.078)
0.50	2.6265	2.6265	2.0669	1.4464	1.6670	1.6670	1.0545	0.7567
	(<0.001)	(9.375)			(<0.001)	(10.984)
1.00	1.8810	1.8810	1.8810	0.9138	1.3532	1.3532	0.6913	0.5160
	(<0.001)	(9.063)			(<0.001)	(11.001)
3.00	1.3324	1.3324	1.3324	0.4992	1.1277	1.1277	0.3795	0.3182
	(<0.001)	(8.750)			(<0.001)	(10.969)

(·) The numerical results in parentheses are computational times in seconds.

). It is noteworthy that although the proposed formula was as accurate as the numerical IE technique for both applications, the computational time using the former was substantially less than using the latter (less than 1 s versus approximately 10 s). Moreover, the ARL₁ results using both techniques were similar to those reported in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7. Similar to the ARL, the SDRL and MRL results also show a decreasing pattern as the shift size was increased for both applications, which is similar to the results in Table 8.

The performance of the proposed formula for both applications running on the modified EWMA control chart is graphically presented in Figure 1. In Figure 1a, the control chart triggered the first out-of-control signal for Application 1 at the sixth point, while in Figure 1b, the control chart detected the first out-of-control signal for Application 2 at the 22nd point.

6. Conclusions

Besides the well-established ARL derivations, more performance techniques adopted for the calculation of the analytical ARL of control charts running underlying autocorrelated processes are needed. A reasonable approach is using the exact formula for the ARL Herein, we provided the derivation of the exact formula for the ARL and an algorithm for calculating its value for an $\mathrm{ARFI}(p, d)$ process running on the modified EWMA control chart. The exact formula was able to sensitively detect a change in the mean of the process. Furthermore, the formula for the out-of-control ARL (ARL₁) is suggested as the best tool for measuring the control chart’s sensitivity to changes in the mean of a process. A performance comparison of the ARL derived by using the exact formula for a process running on the standard EWMA control chart against the modified EWMA control chart shows that the latter was more efficacious. The SDRL and MRL performance measures were found to provide similar results.

Deriving the ARL using the exact formula for an $\mathrm{ARFI}(p, d)$ process running on the modified EWMA control chart is really useful and highly recommended. For long-memory processes, we use the advanced concept of fractional integration when modeling, which has derived from the development of Supharakonsakun et al. [38]. Long-memory processes are found in many fields, such as hydrology or even in economics. An important result of this study is the proposed ARL using the exact formula in this situation which provided the efficacy of the proposed explicit formulas with processes involving real data. In future work, the algorithms for the ARL derived using the exact formula will be implemented as a package freely available for everyone to use. Comparing the modified EWMA control chart with other control charts can also be extended by considering the most desirable control chart parameter settings for $\mathrm{ARFI}(p, d)$ processes in future research.