Analytical Assessment of the DMEWMA Control Chart for Detecting Shifts in ARI and ARFI Models with Applications

Abstract

This paper presents an analytical study of the double modified exponentially weighted moving average (DMEWMA) control chart for monitoring autoregressive integrated (ARI) and autoregressive fractionally integrated (ARFI) processes, with emphasis on its symmetry properties. The explicit formulas for the average run length (ARL) are derived using Fredholm integral equations of the second kind, with existence and uniqueness established via Banach’s fixed-point theorem. The numerical approximations are obtained through the numerical integral equation (NIE) method, and simulations confirm that explicit ARL formulas and the NIE approach yield identical results, validating the theoretical derivations. The results show that symmetry plays a dual role: it ensures performance symmetry in detection across short- and long-memory processes. Comparative studies indicate that for ARI processes, DMEWMA outperforms EWMA and MEWMA for small and moderate shifts, while for ARFI processes, it remains superior to EWMA but shows parity or slight inferiority to MEWMA under certain long-memory conditions. The applications to environmental and economic data illustrate the value of symmetrical control structures in providing robust, unbiased monitoring. With ARL computations completed in under 0.001 s, the DMEWMA chart demonstrates efficiency, balance, and versatility.

1. Introduction

In the contemporary data-driven environment, both environmental and economic systems produce complex time-series signals that require robust and systematic monitoring. For example, per capita plastic-waste generation, which is closely linked to sustainability objectives, and the proportion of government expenditure, which reflects fiscal discipline and efficiency, often exhibit persistence, structural fluctuations, and potential long-memory behavior. Conventional control charts, although foundational, often falter when faced with such intricate dynamics, producing outcomes that lean unevenly toward particular directions or process structures. Against this backdrop, symmetry emerges as more than a mathematical property; it becomes a guiding principle of fairness and harmony in statistical process monitoring. A control chart designed with symmetrical foundations responds in equilibrium to both upward and downward shifts, reflecting changes with mirror-like impartiality. This balanced architecture ensures that detection is not skewed by direction, memory length, or data structure, but instead remains consistent and equitable.

Statistical process control (SPC) provides a comprehensive framework for maintaining quality and identifying departures from expected process behavior [1]. At the foundation of SPC lie control charts, which differentiate between common-cause variation, reflecting the inherent and often symmetrical structure of natural variability, and assignable-cause variation, arising from external disturbances that disrupt this balance [2]. A process is regarded as being in control when variability remains symmetrically distributed around its central tendency, whereas the emergence of asymmetrical or directional deviations is interpreted as evidence of assignable causes and signals an out-of-control state. In this context, the idea of symmetry, understood as balance and orderly distribution, plays a fundamental role in interpreting process behavior. By showing whether fluctuations remain within expected symmetric bounds or deviate in an asymmetric manner, control charts support timely and informed decision-making across diverse fields, including manufacturing, healthcare, environmental monitoring, and finance [3].

Among the earliest and most widely applied schemes is the Shewhart control chart [2]. Although effective for large and abrupt shifts, the Shewhart chart lacks sensitivity to smaller deviations. To address this shortcoming, alternative designs such as the cumulative sum (CUSUM) chart [4] and the exponentially weighted moving average (EWMA) chart were developed. The EWMA chart, first introduced by Roberts [5], is especially valued for its ability to detect small to moderate shifts by incorporating past observations through a smoothing parameter. Over time, extensions and modifications of the EWMA have been proposed to improve sensitivity under various process conditions [6].

The modified EWMA (MEWMA) chart is one such extension, enabling simultaneous monitoring of multiple correlated variables [7]. Research has demonstrated that the MEWMA chart is more effective than the traditional EWMA in capturing process shifts when correlations are present across variables. More recently, further modifications have been proposed to enhance detection power. For example, Patel and Divecha [8] and Khan et al. introduced modified EWMA approaches that demonstrated improved sensitivity to abrupt process shifts. Alevizakos et al. [9] extended this approach to propose the double modified EWMA (DMEWMA), which incorporates an additional smoothing component designed to improve sensitivity to subtle mean shifts. This modification is especially relevant for processes that display gradual or small deviations, where the conventional EWMA or even the MEWMA may lack responsiveness.

Despite these advancements, a persistent limitation is that most existing work assumes independent and identically distributed (i.i.d.) observations. In real-world applications, however, data frequently exhibit autocorrelation and long-range dependence. For instance, economic indicators such as inflation, stock returns, and government spending, as well as environmental measures such as temperature, air quality, and waste generation, often reveal patterns that extend beyond short-term autocorrelation. Traditional autoregressive (AR) and moving average (MA) models have long been used to capture such dependencies [10,11,12], with ARMA models providing a balance between the two. However, when persistence spans longer horizons, fractionally integrated processes are more suitable.

The autoregressive fractionally integrated moving average (ARFIMA) model, introduced by Granger and Joyeux [13] and Hosking [14], incorporates the fractional differencing parameter d to capture long-memory behavior. A restricted version of this model, the autoregressive fractionally integrated (ARFI) process, arises when the moving average component is omitted. The ARFI (p, d) model has proven to be particularly useful for processes where fractional differencing dominates the memory structure [15]. Both the ARI and ARFI processes, therefore, provide a natural framework for analyzing persistent environmental and economic datasets, such as those used in the present study.

Evaluating the performance of control charts in these contexts requires robust metrics. The most widely adopted measure is the average run length (ARL), which quantifies the expected number of samples until a signal is triggered [16]. A high in-control ARL (ARL₀) indicates fewer false alarms, while a low out-of-control ARL (ARL₁) reflects faster detection of actual shifts. Numerous methods have been developed to estimate ARL. Early work applied Markov chain approximations [17] and Monte Carlo simulation [18], while more recent efforts have focused on numerical integral equations (NIE) [19] and explicit formulas [20]. The NIE approach has been especially popular for its generality, whereas explicit formulations are increasingly valued for their computational efficiency and mathematical transparency [21,22]. Nevertheless, relatively few studies have systematically examined ARL behavior for EWMA-type modifications under fractionally integrated processes.

Several contributions in the literature highlight this research gap. Crowder [23] derived analytical results for the ARL of the EWMA charts under Gaussian processes, while Champ and Rigdon [24] examined run-length distributions by comparing Markov chain and the NIE approaches. Subsequent studies have extended explicit formulations to modified EWMA charts, demonstrating their applicability across domains such as manufacturing and finance. Yet, despite these developments, systematic examination of the DMEWMA chart within ARI and ARFI settings remains limited, particularly in terms of empirical evaluation using real world data. In addition, previous studies have paid little attention to the role of symmetry in process behavior. Balanced or proportionate fluctuations typically suggest stability, whereas asymmetrical deviations often imply structural irregularities and potential process deterioration. Addressing this dimension is essential for establishing a more comprehensive understanding of chart performance in complex correlated environments.

The present study addresses this gap by providing an analytical assessment of the DMEWMA control chart for detecting shifts in ARI and ARFI models. Specifically, we derive and validate explicit ARL formulations for the DMEWMA, compare them with the MEWMA and the EWMA benchmarks through simulation, and examine their empirical performance using two real-world datasets: per capita plastic waste and government expenditure. By combining theoretical derivations, simulation analysis, and case studies, this study offers a comprehensive evaluation of the strengths and limitations of DMEWMA in both synthetic and empirical settings. The contributions of this work are threefold: (i) To develop explicit formulas and approximations for the ARL of the DMEWMA chart under ARI and ARFI processes. (ii) To conduct extensive simulations comparing DMEWMA with MEWMA and EWMA, highlighting conditions under which DMEWMA offers superior detection power. (iii) To demonstrate the practical applicability of the approach through two real-world datasets, thereby confirming that the advantages observed in simulations extend to empirical contexts. Through these contributions, this study bridges the gap between theory and practice, showing that advanced EWMA-type charts can effectively monitor processes with long memory and fractional integration. The results are expected to provide valuable insights for both academic research and practical applications in environmental monitoring, economic policy, and beyond.

2. Theoretical Background

2.1. Preliminaries on Autocorrelated and Long-Memory Processes

In modern applications, process data frequently exhibits temporal dependence rather than independence. Autoregressive integrated models use integer differencing to capture short-memory dynamics, whereas autoregressive fractionally integrated models use fractional differencing to reflect long-memory behaviour, which is common in economic, environmental, and engineering data. Such dependencies make monitoring difficult, as standard control charts may produce distorted findings if autocorrelation or persistence are ignored. As a result, developing robust charting approaches that work reliably in both short- and long-memory systems remains an important methodological task. It is worth noting that the ARI and ARFI models together provide a unified framework that covers both short- and long-memory stochastic structures. In particular, when the fractional differencing parameter d = 0 the ARFI process reduces to the conventional ARI (or equivalently, ARMA) model. Therefore, the analytical derivations developed in this study are applicable to a broad class of autocorrelated processes, including ARMA, ARI, and ARFIMA models.

2.1.1. The Autoregressive Integrated

The Autoregressive Integrated model is a simplified version of the ARIMA model in which only the autoregressive and integrated (I) components are considered, excluding the moving average part. This model is suitable when the underlying time series is non-stationary but does not exhibit significant short-term shock effects that would require a moving average component to capture. In other words, the noise component is assumed to be white noise without autocorrelation. The general form of the $ARI(p,d)$ model is:

$${\left(1-B\right)}^d{X}_t=\mu +{\varphi }_1{X}_{t-1}+{\varphi }_2{X}_{t-2}+\dots +{\varphi }_p{X}_{t-p}+{\epsilon }_t,$$

(1)

where: $\mu$ is constant term, ${\varphi }_i$ is autoregressive coefficients, and $\left|{\varphi }_i\right|<1$ at $i=1,2,3,\dots ,p,$ ${\epsilon }_t$ is random error at time $t$ which is ${\epsilon }_t~Exp\left(\beta \right)$, $d$ is degree of differencing to achieve stationarity, and $B$ is backshift operator. The differencing process ${\left(1-B\right)}^d$ is applied first to remove trends or non-stationarity, followed by modeling the differenced series using an $AR\left(p\right)$ structure.

2.1.2. The Autoregressive Fractionally Integrated

The Autoregressive Fractionally Integrated model is a generalization of the ARI model that allows for fractional differencing, rather than restricting to integer orders of differencing. This is particularly useful for modelling long-memory processes, where dependencies between observations decay slowly over time. The ARFI model can capture both short-term dynamics (via AR terms) and long-term dependencies (via fractional differencing). The general form of the $ARFI\left(p,d\right)$ model is:

$${\left(1-B\right)}^d{X}_t=\mu +{\varphi }_1{X}_{t-1}+{\varphi }_2{X}_{t-2}+\dots +{\varphi }_p{X}_{t-p}+{\epsilon }_t, 0<d<1,$$

(2)

where: $d$ is fractional differencing parameter (not necessarily an integer), all other terms are as previously defined. Fractional differencing is defined using the binomial expansion: ${\left(1-B\right)}^d={\displaystyle \sum _{k=0}^{\infty }\frac{\Gamma \left(k-d\right)}{\Gamma \left(k+1\right)\Gamma \left(-d\right)}}{B}^k.$ This model is particularly suitable for time series with long-range dependence that are not adequately addressed by standard ARIMA models.

2.2. Average Run Length as a Performance Measure

The average run length is the most commonly used performance indicator when assessing control charts. It is defined as the estimated number of samples required before a signal is generated. Two instances are considered: the in-control ARL, indicating the average time to a false alarm, and the out-of-control ARL, reflecting the average time to detect an actual shift. An effective chart has a large ARL₀ to prevent false alarms and a small ARL₁ to notice changes quickly. Because of its dual purpose in balancing false-alarm prevention and detection power, ARL is used as a standard criterion in both theoretical and applied evaluations of control chart performance.

2.3. Control Charting Schemes Considered in This Study

A sequential statistical tool for tracking process stability and identifying deviations from an out-of-control state is a control chart. Three fundamental elements make up the generic structure of a control chart: (i) a monitoring statistic that is based on observed data; (ii) decision thresholds called upper and lower control limits; and (iii) a signalling rule that sounds an alarm when the statistic beyond the designated bounds. Shewhart-type charts have historically been used to identify significant and sudden changes, but their sensitivity to subtler or more gradual changes is constrained. Memory-type control charts, such the exponentially weighted moving average and its variations, use past data to work around this restriction and improve response to mild process changes.

2.3.1. The Traditional Exponentially Weighted Moving Average (EWMA)

Roberts [5] introduced the EWMA chart, which is one of the most used memory-type control charts. Its test statistic is calculated by applying an exponential weighting system to previous observations, with more recent values receiving greater weight. This cumulative structure enables EWMA to respond more effectively to minor and moderate changes than Shewhart-type charts. However, its performance may degrade when applied to processes with high autocorrelation or long-memory features, necessitating additional changes. The EWMA statistic $E_t$ is computed recursively:

$$E_t=\lambda X_t+\left(1-\lambda \right)E_{t-1}, 0<\lambda \le 1,$$

(3)

where $X_t$ is the observed value at time t, $E_{t-1}$ is previous EWMA value, $\lambda$ is the smoothing constant (typically $0<\lambda \le 1$). The initial value $E_0$ is often set to the process target mean ${\mu }_0$ [25]. The mean value and the variance in the EWMA control chart are given by:

$$\begin{gathered} E\left(E_t\right)={\mu }_0 \\ Var\left(E_t\right)={\displaystyle \frac{\lambda {\sigma }^2}{\left(2-\lambda \right)}}. \end{gathered}$$

For the control limits, the center line (CL), upper control limit (UCL) and lower control limit (LCL) are given by the following formulas:

$$\begin{gathered} UC{L}_t={\mu }_0+L_E\sigma \sqrt{{\displaystyle \frac{\lambda }{2-\lambda }}}, \\ CL={\mu }_0, \\ LC{L}_t={\mu }_0-L_E\sigma \sqrt{{\displaystyle \frac{\lambda }{2-\lambda }}}. \end{gathered}$$

(4)

where $L_E>0$ is the control limit width (usually 2.5–3.0), and $\sigma$ is the process standard deviation.

2.3.2. The Modified Exponentially Weighted Moving Average (MEWMA)

The MEWMA chart expands on the traditional EWMA framework by changing the weighting process and, in some situations, incorporating a multivariate adaption. This improvement enhances detection capabilities in correlated or structured data, which is useful in cases where process persistence is significant. Previous research has demonstrated that MEWMA can be competitive in fractionally integrated processes because its structure allows for greater responsiveness to long-memory dependencies. Nonetheless, MEWMA may not consistently outperform competing schemes for all shift sizes and process types. The formula of improved MEWMA control chart is defined as

$$M_t=\lambda X_t+(1-\lambda )M_{t-1}+c(X_t-X_{t-1}), 0<\lambda \le 1,$$

(5)

where $X_t$ is the observed value at time t, $M_{t-1}$ is previous MEWMA value, $\lambda$ is the smoothing constant (typically $0<\lambda \le 1$). $c$ is an additional traditional parameter, in case $c=1$ it become to the original MEWMA control chart [26], and in case $c=0$ it become to the EWMA control chart. The mean value and the variance in the MEWMA control chart are given by:

$$\begin{gathered} E\left(M_t\right)={\mu }_0 \\ Var\left(M_t\right)=\left[{\displaystyle \frac{\left(\lambda +2\lambda c+2c^2\right)-\lambda {\left(1-\lambda -c\right)}^2{\left(1-\lambda \right)}^{2t-2}}{2-\lambda }}\right]{\sigma }^2. \end{gathered}$$

For the control limits, the center line (CL), upper control limit (UCL) and lower control limit (LCL) are given by the following formulas:

$$\begin{array}{c}UC{L}_t={\mu }_0+L_M\sigma \sqrt{{\displaystyle \frac{\left(\lambda +2\lambda c+2c^2\right)}{2-\lambda }}},\\ CL={\mu }_0,\\ LC{L}_t={\mu }_0-L_M\sigma \sqrt{{\displaystyle \frac{\left(\lambda +2\lambda c+2c^2\right)}{2-\lambda }}},\end{array}$$

(6)

where $L_M>0$ is the control limit width (usually 2.5–3.0), and $\sigma$ is the process standard deviation.

2.3.3. The Double Modified Exponentially Weighted Moving Average (DMEWMA)

By applying a double modification to the EWMA structure, the DMEWMA chart improves its capacity to identify process shifts under a variety of circumstances. While maintaining resilience against parameter fluctuation, the extra layer of modification increases sensitivity to minor and moderate changes. The system of equations defined by

$$\begin{array}{c}{M}_t={\lambda }_1{X}_t+(1-{\lambda }_1)M_{t-1}+c_1(X_t-X_{t-1})\\ D_t={\lambda }_2{M}_t+(1-{\lambda }_2)D_{t-1}+c_2(M_t-M_{t-1})\end{array},$$

(7)

where $X_t$ is the observed value at time t, $D_{t-1}$ is previous DMEWMA value, ${\lambda }_1,{\lambda }_2$ are the smoothing constants such that $0<{\lambda }_1,{\lambda }_2\le 1$, $c_1,c_2$ are the additional traditional parameters. In general, the results indicate that moderate values of ${\lambda }_1\in \left[0.1,0.3\right]$ and ${\lambda }_2\in \left[0.05,0.15\right]$ combined with modification coefficients $c_1,c_2\in \left[0.5,1.5\right]$ provide a stable balance between sensitivity to small shifts and robustness to autocorrelation. A summary of the recommended parameter combinations for achieving target ARL₀ values (370, and 500) is provided. The initial value $X_0=M_0=D_0$ are often set to the process target mean ${\mu }_0$. The mean value and the variance in the statistic $D_t$ are given by:

$$E\left(D_t\right)={\mu }_0$$

$$\begin{gathered} {\displaystyle \frac{Var\left(D_t\right)}{{\sigma }^2}={\left(c+\lambda \right)}^4+4{\lambda }^2{\left(c+\lambda \right)}^2{\left(c+\lambda -1\right)}^2+\frac{4{\lambda }^2{\left(c+\lambda \right)}^2{\left(c+\lambda -1\right)}^2{\left(1-\lambda \right)}^2}{1-{\left(1-\lambda \right)}^2} \\ -\frac{4{\lambda }^3\left(c+\lambda \right){\left(c+\lambda -1\right)}^3\left(1-\lambda \right)}{{\left(1-{\left(1-\lambda \right)}^2\right)}^2}+{\lambda }^4{\left(c+\lambda -1\right)}^4\frac{1+{\left(1-\lambda \right)}^2}{{\left(1-{\left(1-\lambda \right)}^2\right)}^3}.} \end{gathered}$$

For the control limits, the center line (CL), upper control limit (UCL) and lower control limit (LCL) are given by the following formulas:

$$\begin{array}{c}UC{L}_t={\mu }_0+L_D\sigma \sqrt{Var\left(D_t\right)},\\ CL={\mu }_0,\\ LC{L}_t={\mu }_0-L_D\sigma \sqrt{Var\left(D_t\right)},\end{array}$$

(8)

where $L_D>0$ is the control limit width (usually 2.5–3.0), and $\sigma$ is the process standard deviation.

3. Analytical Derivation and Approximate of ARL for the DMEWMA Control Chart

In this section, the derivation of the ARL is developed through two distinct approaches: an explicit analytical expression and the NIE technique, both established within the ARI(p,d) and ARFI(p,d) modelling structures.

3.1. Analysis of the Average Run Length Within the ARI(p,d) Model

From Equation (7) can be modified by substituting the $ARI\left(p,d\right)$ process $X_t$ from Equation (1). The recursive formula of DMEWMA statistics $D_t$ can be written as follows:

$$D_t=\left(1-{\lambda }_2\right)D_{t-1}+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_{t-1}+\left({\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2\right)Z_t-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_{t-1}.$$

Let $Z_t$ be the observation on $ARI\left(p,d\right)$ model, then:

$$\begin{array}{ll}{D}_t=& \left(1-{\lambda }_2\right)D_{t-1}+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_{t-1}-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_{t-1}+\left({\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2\right)\times \\ & \left[\begin{array}{l}\mu +{\epsilon }_t+\left(d{Z}_{t-1}-{\displaystyle \frac{d\left(d-1\right)}{2!}}{Z}_{t-2}+{\displaystyle \frac{d\left(d-1\right)\left(d-2\right)}{3!}}{Z}_{t-3}-\dots \right)\\ +{\displaystyle \sum _{i=1}^p{\varphi }_i\left(Z_{t-i}-d{Z}_{t-\left(i+1\right)}+\frac{d\left(d-1\right)}{2!}{Z}_{t-\left(i+2\right)}-\dots \right)}\end{array}\right],\end{array}$$

Set $D_{t-1}=s,$ $M_{t-1}=M_0,$ $Z_{t-1}=Z_0,$ $H={\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2,$ and

$$\begin{array}{l}N=\mu +\left(d{Z}_{t-1}-{\displaystyle \frac{d\left(d-1\right)}{2!}}{Z}_{t-2}+{\displaystyle \frac{d\left(d-1\right)\left(d-2\right)}{3!}}{Z}_{t-3}-\dots \right)\\ \begin{array}{ccc}& & +\end{array}{\displaystyle \sum _{i=1}^p{\varphi }_i\left(Z_{t-i}-d{Z}_{t-\left(i+1\right)}+\frac{d\left(d-1\right)}{2!}{Z}_{t-\left(i+2\right)}-\dots \right)}.\end{array}$$

Then

$$D_t=\left(1-{\lambda }_2\right)s+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0+H{\epsilon }_t+HN.$$

(9)

The stopping time, denoted as $\tau ,$ is a random variable representing the time (or number of observations) until the DMEWMA statistic first exceeds the control limits, signaling a process change. Mathematically, it can be defined as follows:

$${\tau }_{LCL,UCL}=\inf \left\{t>0;D_t\notin \left(LCL,UCL\right)\right\},$$

where $LCL$ is the lower control limit, and $UCL$ is the upper control limit.

Consider a one-sided case for the in-control process, where $0\le D_t<h,$ with $LCL=0$ and $UCL=h$. So that:

$$0\le \left(1-{\lambda }_2\right)s+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0+H{\epsilon }_t+HN<h.$$

Next, if rearranged into the form of ${\epsilon }_t$, we get:

$$\begin{array}{l}{\displaystyle \frac{-\left(1-{\lambda }_2\right)s-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0}{H}}-N\begin{array}{ccc}\le & {\epsilon }_t& <\end{array}\\ {\displaystyle \frac{h-\left(1-{\lambda }_2\right)s-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0}{H}}-N.\end{array}$$

(10)

3.1.1. Explicit Analytical Formulation

The explicit formulas for the ARL of the DMEWMA control chart, derived for observations from autocorrelated processes with exponential white noise, are obtained through a Fredholm integral equation of the second kind. Subsequently, the integral equations are rigorously established to ensure the existence and uniqueness of their solutions. The explicit derivations can be found in Appendix A, whereas the proofs of existence and uniqueness are detailed in Appendix B. The explicit formula for the ARL of $ARI\left(p,d\right)$ process on the DMEWMA control chart as

$$\Pi \left(s\right)=1+\left[{\displaystyle \frac{e^{\frac{\left(1-{\lambda }_2\right)s}{\beta H}}\left(1-e^{\frac{-h}{\beta H}}\right)}{\left(\frac{e^{\frac{-{\lambda }_{2}h}{\beta H}}-1}{{\lambda }_2}\right)+e^{\frac{-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0-HN}{\beta H}}}}\right].$$

(11)

For the ARL solution of an in-control process, we set $AR{L}_0=370,500$ when the exponential parameter is defined to $\beta ={\beta }_0$ and for an out-of-control we defined $AR{L}_1$ which is $\beta ={\beta }_1={\beta }_0\left(1+\delta \right)$ where $\delta$ is the shift size.

3.1.2. Numerical Integral Equation (NIE) Approach

The ARL is often estimated using the NIE techniques. Common quadrature rules, including the midpoint, trapezoidal, Simpson, and Gauss–Legendre methods, are typically employed to approximate the associated integral equation. In this study, the DMEWMA statistic for various processes is evaluated using the midpoint rule. For the approximation at arbitrary points,

$${\Pi }_N\left(u\right)=1+{\displaystyle \frac{h}{H}}{\displaystyle \sum _{j=1}^m\Pi \left(x_j\right)}f\left({\displaystyle \frac{x_j-\left(1-{\lambda }_2\right)s_i-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0-HN}{H}}\right),$$

(12)

The procedure for which is presented in Appendix C.

3.2. Analysis of the Average Run Length Within the ARFI(p,d) Model

Equation (7) can be modified by substituting the $ARFI\left(p,d\right)$ process $X_t$ from Equation (2). The recursive formula of DMEWMA statistics $D_t$ can be written as follows:

$$D_t=\left(1-{\lambda }_2\right)D_{t-1}+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_{t-1}+\left({\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2\right)Z_t-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_{t-1}.$$

Let $Z_t$ be the observation on $ARFI\left(p,d\right)$ model, then:

$$\begin{array}{ll}{D}_t =& \left(1-{\lambda }_2\right)D_{t-1}+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_{t-1}\left(c_1{\lambda }_2+c_1{c}_2\right)Z_{t-1}+\left({\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2\right)\times \\ & \left[\begin{array}{l}\mu +{\epsilon }_t+\left(d{Z}_{t-1}-{\displaystyle \frac{d\left(d-1\right)}{2!}}{Z}_{t-2}+{\displaystyle \frac{d\left(d-1\right)\left(d-2\right)}{3!}}{Z}_{t-3}-\dots \right)\\ +{\displaystyle \sum _{i=1}^p{\varphi }_i\left(Z_{t-i}-d{Z}_{t-\left(i+1\right)}+\frac{d\left(d-1\right)}{2!}{Z}_{t-\left(i+2\right)}-\dots \right)}\end{array}\right], 0<d<1.\end{array}$$

Set $D_{t-1}=s,$ $M_{t-1}=M_0,$ $Z_{t-1}=Z_0,$ $H={\lambda }_1{\lambda }_2+c_1{\lambda }_2+c_2{\lambda }_1+c_1{c}_2,$ and

$$\begin{array}{l}N=\mu +\left(d{Z}_{t-1}-{\displaystyle \frac{d\left(d-1\right)}{2!}}{Z}_{t-2}+{\displaystyle \frac{d\left(d-1\right)\left(d-2\right)}{3!}}{Z}_{t-3}-\dots \right)\\ \begin{array}{ccc}& & +\end{array}{\displaystyle \sum _{i=1}^p{\varphi }_i\left(Z_{t-i}-d{Z}_{t-\left(i+1\right)}+\frac{d\left(d-1\right)}{2!}{Z}_{t-\left(i+2\right)}-\dots \right)}.\end{array}$$

Then

$$D_t=\left(1-{\lambda }_2\right)s+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0+H{\epsilon }_t+HN.$$

(13)

The stopping time, denoted as $\tau ,$ is a random variable representing the time (or number of observations) until the DMEWMA statistic first exceeds the control limits, signaling a process change. Mathematically, it can be defined as follows:

$${\tau }_{LCL,UCL}=\inf \left\{t>0;D_t\notin \left(LCL,UCL\right)\right\},$$

where $LCL$ is the lower control limit, and $UCL$ is the upper control limit.

Consider a one-sided case for the in-control process, where $0\le D_t<h,$ with $LCL=0$ and $UCL=h$. So that,

$$0\le \left(1-{\lambda }_2\right)s+\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0-\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0+H{\epsilon }_t+HN<h.$$

Next, if rearranged into the form of ${\epsilon }_t$, we get:

$$\begin{array}{l}{\displaystyle \frac{-\left(1-{\lambda }_2\right)s-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0}{H}}-N\begin{array}{ccc}\le & {\epsilon }_t& <\end{array}\\ {\displaystyle \frac{h-\left(1-{\lambda }_2\right)s-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0}{H}}-N.\end{array}$$

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3.2.1. Explicit Analytical Formulation

The explicit formulas for the ARL of the DMEWMA control chart, derived for observations from autocorrelated processes with exponential white noise, are obtained through a Fredholm integral equation of the second kind. Subsequently, the integral equations are rigorously established to ensure the existence and uniqueness of their solutions. The explicit formula for the ARL of $ARFI\left(p,d\right)$ process on the DMEWMA control chart as follows:

$$\mathsf{\Xi }\left(s\right)=1+\left[{\displaystyle \frac{e^{\frac{\left(1-{\lambda }_2\right)s}{\beta H}}\left(1-e^{\frac{-h}{\beta H}}\right)}{\left(\frac{e^{\frac{-{\lambda }_{2}h}{\beta H}}-1}{{\lambda }_2}\right)+e^{\frac{-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0-HN}{\beta H}}}}\right].$$

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For the ARL solution of an in-control process, we set $AR{L}_0=370,500$ when the exponential parameter is defined to $\beta ={\beta }_0$ and for an out-of-control we defined $AR{L}_1$ which is $\beta ={\beta }_1={\beta }_0\left(1+\delta \right)$ where $\delta$ is the shift size.

3.2.2. Numerical Integral Equation (NIE) Approach

The ARL has frequently been estimated using the NIE technique. The most common and widely used quadrature rules—midpoint rule, trapezoidal rule, Simpson’s rule, and Gauss-Legendre rule—can be used to approximate the integral equation. In this study, the DMEWMA statistic for several processes was subjected to midpoint rule. For the Approximation for arbitrary points,

$${\mathsf{\Xi }}_N\left(u\right)=1+{\displaystyle \frac{h}{H}}{\displaystyle \sum _{j=1}^m\mathsf{\Xi }\left(x_j\right)}f\left({\displaystyle \frac{x_j-\left(1-{\lambda }_2\right)s_i-\left({\lambda }_2-{\lambda }_1{\lambda }_2-{\lambda }_1{c}_1\right)M_0+\left(c_1{\lambda }_2+c_1{c}_2\right)Z_0-HN}{H}}\right),$$

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4. Simulation Design

The initial goal of the simulation study is to evaluate the performance of the explicit ARL formulas developed in Section 4 to the numerical integral equation approach. Although both systems are predicted to provide the same ARL results, their computing efficiency varies significantly. Demonstrating this comparability while emphasizing the efficiency improvements of the explicit approach is critical for determining its practical relevance. The second goal is to assess and compare the performance of three control charting schemes—EWMA, MEWMA, and DMEWMA—for both short-memory ARI and long-memory ARFI processes. The comparison focuses on each charts sensitivity to minor and moderate changes in process mean, as well as its overall detection capability as evaluated by the relative mean index (RMI). This two-part analysis provides a fair evaluation of methodological soundness and comparative effectiveness across various process topologies.

For the EWMA chart, the smoothing parameter λ was varied between 0.05 and 0.40, consistent with conventional practice. For each λ, the control limit constant was adjusted until the in-control average run length (ARL₀) was approximately 370, matching the standard benchmark for Shewhart-type charts. For the MEWMA chart, two parameters (λ, c) were considered. The same ARL₀-matching procedure was applied by simulating multiple (λ, c) pairs and retaining those producing ARL₀ ≈ 370. The ccc parameter, which governs the contribution of lagged differences, was tuned in the range of 0.05–0.50 to ensure stability. For the proposed DMEWMA chart, four parameters (λ₁, λ₂, c₁, c₂) were explored within the same ranges as above. To simplify computation and practical implementation, symmetry was assumed (c₁ = c₂ = c), following Patel and Divecha (2011) [8]. Monte Carlo simulations were performed to estimate ARL₀, and the control limits were iteratively adjusted until ARL₀ ≈ 370, consistent with EWMA and MEWMA. Once equal in-control performance was established (ARL₀ ≈ 370 for all charts), out-of-control ARL (ARL₁) values were compared across various mean shifts (0.5σ–2.0σ). The final parameter sets were selected to achieve minimum ARL₁ under the maintained ARL₀ condition, ensuring a fair and optimized comparison.

For all simulations, the in-control ARL₀ is designed to be approximately 370, and 500, following standard practice in quality control applications. Out-of-control performance (ARL₁) is examined under a wide range of mean shift sizes, including $\delta =0.01,0.05,0.1,0.5,1$ and $3.$ The smoothing constant $\lambda$ is varied to reflect both low and moderate levels of memory, and the modification parameters of MEWMA and DMEWMA are tuned according to conventional design strategies. Each scenario is replicated sufficiently to obtain stable ARL estimates, and CPU times are recorded for both the explicit formulas and the NIE method to assess computational efficiency.

4.1. Performance of the Control Charts

This study evaluates the performance of two methods for calculating the Average Run Length: the explicit formulas and the Numerical Integral Equation approach. Computations were conducted in Mathematica, employing a sufficient number of division points to ensure numerical precision.

Table 1. Comparison of the accuracy of the exact formula and NIE technique for the $ARI(1,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.05,0.1\right)$.

Coefficients		$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	1	0.00	370.050 (<0.001)	370.050 (3.70)	100	369.678 (<0.001)	369.678 (3.65)	100	370.000 (<0.001)	370.000 (3.45)	100
		0.01	249.902 (<0.001)	249.902 (3.17)	100	104.437 (<0.001)	104.437 (3.73)	100	12.887 (<0.001)	12.887 (3.75)	100
		0.05	104.974 (<0.001)	104.974 (3.78)	100	27.071 (<0.001)	27.071 (3.48)	100	3.361 (<0.001)	3.361 (3.23)	100
		0.10	58.302 (<0.001)	58.302 (3.38)	100	14.147 (<0.001)	14.147 (3.26)	100	2.140 (<0.001)	2.140 (3.84)	100
		0.50	9.851 (<0.001)	9.851 (3.56)	100	3.299 (<0.001)	3.299 (3.78)	100	1.184 (<0.001)	1.184 (3.91)	100
		1.00	4.279 (<0.001)	4.279 (3.98)	100	2.025 (<0.001)	2.025 (3.15)	100	1.079 (<0.001)	1.079 (3.33)	100
		3.00	1.651 (<0.001)	1.651 (3.11)	100	1.285 (<0.001)	1.285 (3.26)	100	1.021 (<0.001)	1.021 (3.57)	100
	2	0.00	369.892 (<0.001)	369.892 (3.46)	100	369.678 (<0.001)	369.678 (3.42)	100	370.309 (<0.001)	370.309 (3.71)	100
		0.01	249.828 (<0.001)	249.828 (3.77)	100	104.437 (<0.001)	104.437 (3.37)	100	12.887 (<0.001)	12.887 (3.24)	100
		0.05	104.960 (<0.001)	104.960 (3.24)	100	27.071 (<0.001)	27.071 (3.99)	100	3.361 (<0.001)	3.361 (3.57)	100
		0.10	58.297 (<0.001)	58.297 (3.40)	100	14.147 (<0.001)	14.147 (3.76)	100	2.140 (<0.001)	2.140 (3.81)	100
		0.50	9.851 (<0.001)	58.297 (3.56)	100	3.299 (<0.001)	3.299 (3.82)	100	1.184 (<0.001)	1.184 (3.62)	100
		1.00	4.279 (<0.001)	4.279 (3.12)	100	2.025 (<0.001)	2.025 (3.15)	100	1.079 (<0.001)	1.079 (3.18)	100
		3.00	1.651 (<0.001)	1.651 (3.29)	100	1.285 (<0.001)	1.285 (3.55)	100	1.021 (<0.001)	1.021 (3.34)	100

,

Table 2. Comparison of the accuracy of the exact formula and NIE technique for the $ARI(1,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.2,0.25\right)$.

Coefficients		$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	1	0.00	370.114 (<0.001)	370.114 (3.11)	100	370.097 (<0.001)	370.097 (3.23)	100	369.992 (<0.001)	369.992 (3.14)	100
		0.01	177.661 (<0.001)	177.661 (3.56)	100	78.975 (<0.001)	78.975 (3.87)	100	11.129 (<0.001)	11.129 (3.88)	100
		0.05	55.795 (<0.001)	55.795 (3.31)	100	19.288 (<0.001)	19.288 (3.44)	100	3.010 (<0.001)	3.010 (3.31)	100
		0.10	28.974 (<0.001)	28.974 (3.78)	100	10.092 (<0.001)	10.092 (3.98)	100	1.973 (<0.001)	1.973 (3.26)	100
		0.50	5.113 (<0.001)	5.113 (3.59)	100	2.545 (<0.001)	2.545 (3.53)	100	1.160 (<0.001)	1.160 (3.69)	100
		1.00	2.549 (<0.001)	2.549 (3.91)	100	1.678 (<0.001)	1.678 (3.30)	100	1.069 (<0.001)	1.069 (3.84)	100
		3.00	1.319 (<0.001)	1.319 (3.26)	100	1.183 (<0.001)	1.183 (3.22)	100	1.019 (<0.001)	1.019 (3.21)	100
	2	0.00	370.114 (<0.001)	370.114 (3.93)	100	370.097 (<0.001)	370.097 (3.86)	100	369.992 (<0.001)	369.992 (3.74)	100
		0.01	177.661 (<0.001)	177.661 (3.27)	100	78.975 (<0.001)	78.975 (3.97)	100	11.129 (<0.001)	11.129 (3.95)	100
		0.05	55.795 (<0.001)	55.795 (3.31)	100	19.288 (<0.001)	19.288 (3.15)	100	3.010 (<0.001)	3.010 (3.84)	100
		0.10	28.974 (<0.001)	28.974 (3.57)	100	10.092 (<0.001)	10.092 (3.79)	100	1.973 (<0.001)	1.973 (3.05)	100
		0.50	5.113 (<0.001)	5.113 (3.59)	100	2.545 (<0.001)	2.545 (3.39)	100	1.160 (<0.001)	1.160 (2.77)	100
		1.00	2.549 (<0.001)	2.549 (3.40)	100	1.678 (<0.001)	1.678 (3.20)	100	1.069 (<0.001)	1.069 (3.65)	100
		3.00	1.319 (<0.001)	1.319 (3.32)	100	1.183 (<0.001)	1.183 (3.24)	100	1.019 (<0.001)	1.019 (3.82)	100

,

Table 3. Comparison of the accuracy of the exact formula and NIE technique for the $ARI(2,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.05,0.1\right)$.

Coefficients			$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.2	1	0.00	369.776 (<0.001)	369.776 (3.99)	100	369.896 (<0.001)	369.896 (4.13)	100	370.330 (<0.001)	370.330 (4.27)	100
			0.01	252.129 (<0.001)	252.129 (4.29)	100	102.022 (<0.001)	102.022 (4.22)	100	11.872 (<0.001)	11.872 (4.27)	100
			0.05	106.675 (<0.001)	106.675 (4.17)	100	26.168 (<0.001)	26.168 (4.25)	100	3.148 (<0.001)	3.148 (4.41)	100
			0.10	59.047 (<0.001)	59.047 (4.54)	100	13.603 (<0.001)	13.603 (4.21)	100	2.033 (<0.001)	2.033 (4.36)	100
			0.50	9.573 (<0.001)	9.573 (4.97)	100	3.124 (<0.001)	3.124 (4.39)	100	1.162 (<0.001)	1.162 (4.39)	100
			1.00	4.055 (<0.001)	4.055 (4.26)	100	1.921 (<0.001)	1.921 (4.63)	100	1.068 (<0.001)	1.068 (4.48)	100
			3.00	1.567 (<0.001)	1.567 (4.77)	100	1.244 (<0.001)	1.244 (4.63)	100	1.017 (<0.001)	1.017 (4.58)	100
		2	0.00	370.588 (<0.001)	370.588 (4.25)	100	369.896 (<0.001)	369.896 (4.76)	100	370.330 (<0.001)	370.330 (4.28)	100
			0.01	252.518 (<0.001)	252.518 (4.54)	100	102.022 (<0.001)	102.022 (4.93)	100	11.872 (<0.001)	11.872 (4.37)	100
			0.05	106.753 (<0.001)	106.753 (4.32)	100	26.168 (<0.001)	26.168 (4.03)	100	3.148 (<0.001)	3.148 (4.64)	100
			0.10	59.075 (<0.001)	59.075 (4.48)	100	13.603 (<0.001)	13.603 (4.38)	100	2.033 (<0.001)	2.033 (4.59)	100
			0.50	9.575 (<0.001)	9.575 (4.67)	100	3.124 (<0.001)	3.124 (4.79)	100	1.162 (<0.001)	1.162 (4.63)	100
			1.00	4.055 (<0.001)	4.055 (4.76)	100	1.921 (<0.001)	1.921 (4.47)	100	1.068 (<0.001)	1.068 (4.78)	100
			3.00	1.567 (<0.001)	1.567 (4.87)	100	1.244 (<0.001)	1.244 (4.68)	100	1.017 (<0.001)	1.017 (4.90)	100

and

Table 4. Comparison of the accuracy of the exact formula and NIE technique for the $ARI(2,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.2,0.25\right)$.

Coefficients			$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.2	1	0.00	369.964 (<0.001)	369.964 (4.32)	100	369.885 (<0.001)	369.885 (4.53)	100	369.935 (<0.001)	369.935 (4.53)	100
			0.01	176.976 (<0.001)	176.976 (4.31)	100	75.763 (<0.001)	75.763 (4.39)	100	10.191 (<0.001)	10.191 (4.59)	100
			0.05	55.219 (<0.001)	55.219 (4.29)	100	18.317 (<0.001)	18.317 (4.40)	100	2.814 (<0.001)	2.814 (4.61)	100
			0.10	28.496 (<0.001)	28.496 (4.48)	100	9.554 (<0.001)	9.554 (4.65)	100	1.875 (<0.001)	1.875 (4.53)	100
			0.50	4.882 (<0.001)	4.882 (4.56)	100	2.407 (<0.001)	2.407 (4.39)	100	1.140 (<0.001)	1.140 (4.42)	100
			1.00	2.415 (<0.001)	2.415 (4.63)	100	1.602 (<0.001)	1.602 (4.55)	100	1.060 (<0.001)	1.060 (4.62)	100
			3.00	1.277 (<0.001)	1.277 (4.47)	100	1.156 (<0.001)	1.156 (4.56)	100	1.016 (<0.001)	1.016 (4.64)	100
		2	0.00	369.964 (<0.001)	369.964 (4.35)	100	369.885 (<0.001)	369.885 (4.77)	100	369.935 (<0.001)	369.935 (4.50)	100
			0.01	176.976 (<0.001)	176.976 (4.36)	100	75.763 (<0.001)	75.763 (4.39)	100	10.191 (<0.001)	10.191 (4.44)	100
			0.05	55.219 (<0.001)	55.219 (4.35)	100	18.317 (<0.001)	18.317 (4.47)	100	2.814 (<0.001)	2.814 (4.45)	100
			0.10	28.496 (<0.001)	28.496 (4.42)	100	9.554 (<0.001)	9.554 (4.72)	100	1.875 (<0.001)	1.875 (4.81)	100
			0.50	4.882 (<0.001)	4.882 (4.51)	100	2.407 (<0.001)	2.407 (4.80)	100	1.140 (<0.001)	1.140 (4.53)	100
			1.00	2.415 (<0.001)	2.415 (4.60)	100	1.602 (<0.001)	1.602 (4.79)	100	1.060 (<0.001)	1.060 (4.47)	100
			3.00	1.277 (<0.001)	1.277 (4.72)	100	1.156 (<0.001)	1.156 (4.66)	100	1.016 (<0.001)	1.016 (4.45)	100

compare the Exact analytical method with the Numerical Integral Equation technique for ARL evaluation in ARI and ARFI processes. Both methods yield numerically identical results with 100% accuracy, confirming the validity of the Exact formulations. However, the Exact method computes results in less than 0.001 s, whereas the NIE method requires 3–5 s depending on parameter settings. This efficiency advantage makes the Exact approach particularly suitable for real time monitoring and large-scale simulations.

Following standard statistical process control practice, an in-control ARL (ARL₀) of 370 was adopted, where lower ARL values indicate increased sensitivity in detecting process shifts, as reflected in the out-of-control ARL (ARL₁) measurements. For each process type, the explicit solutions were computed using derived analytical expressions, while the NIE method implementations utilized corresponding numerical formulations.

To quantify the agreement between methods, we employed the Accuracy percentage criteria and CPU times:

$$\%ACC=\left|{\displaystyle \frac{H\left(\xi \right)-\tilde{H}\left(\xi \right)}{H\left(\xi \right)}}\right|\times 100\%$$

where $H(\xi )$ and $\tilde{H}(\xi )$ are the ARL from the explicit formula, and NIE method, respectively.

The correctness of the exact formulations under both ARFI(1,d) and ARFI(2,d) procedures is confirmed by

Table 5. Comparison of the accuracy of the exact formula and NIE technique for the $ARFI(1,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.05,0.1\right)$.

Coefficients		$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.15	0.00	370.260 (<0.001)	370.260 (4.30)	100	370.011 (<0.001)	370.011 (4.57)	100	369.972 (<0.001)	369.972 (4.44)	100
		0.01	323.245 (<0.001)	323.245 (4.15)	100	328.679 (<0.001)	328.679 (4.37)	100	313.327 (<0.001)	313.327 (4.55)	100
		0.05	192.664 (<0.001)	192.664 (4.17)	100	209.304 (<0.001)	209.304 (4.71)	100	170.614 (<0.001)	170.614 (4.25)	100
		0.10	106.404 (<0.001)	106.404 (4.38)	100	124.707 (<0.001)	124.707 (4.46)	100	88.176 (<0.001)	88.176 (3.46)	100
		0.50	4.540 (<0.001)	4.540 (4.29)	100	7.449 (<0.001)	7.449 (4.27)	100	3.857 (<0.001)	3.857 (4.28)	100
		1.00	1.320 (<0.001)	1.320 (4.40)	100	1.794 (<0.001)	1.794 (4.83)	100	1.298 (<0.001)	1.298 (4.39)	100
		3.00	1.007 (<0.001)	1.007 (4.28)	100	1.027 (<0.001)	1.027 (4.63)	100	1.009 (<0.001)	1.009 (4.20)	100
	0.30	0.00	369.913 (<0.001)	369.913 (4.57)	100	369.927 (<0.001)	369.927 (4.23)	100	370.067 (<0.001)	370.067 (4.19)	100
		0.01	322.132 (<0.001)	322.132 (4.26)	100	327.612 (<0.001)	327.612 (4.42)	100	309.753 (<0.001)	309.753 (4.43)	100
		0.05	190.183 (<0.001)	190.183 (4.43)	100	206.282 (<0.001)	206.282 (4.37)	100	162.907 (<0.001)	162.907 (4.22)	100
		0.10	103.908 (<0.001)	103.908 (4.30)	100	121.378 (<0.001)	121.378 (4.72)	100	81.909 (<0.001)	81.909 (3.64)	100
		0.50	4.252 (<0.001)	4.252 (4.29)	100	6.878 (<0.001)	6.878 (4.38)	100	3.452 (<0.001)	3.452 (4.26)	100
		1.00	1.282 (<0.001)	1.282 (4.45)	100	1.694 (<0.001)	1.694 (4.33)	100	1.248 (<0.001)	1.248 (4.37)	100
		3.00	1.006 (<0.001)	1.006 (4.65)	100	1.022 (<0.001)	1.022 (4.39)	100	1.007 (<0.001)	1.007 (4.45)	100
	0.45	0.00	370.060 (<0.001)	370.060 (4.14)	100	370.049 (<0.001)	370.049 (4.16)	100	369.944 (<0.001)	369.944 (4.50)	100
		0.01	321.727 (<0.001)	321.727 (4.44)	100	327.066 (<0.001)	327.066 (4.27)	100	307.294 (<0.001)	307.294 (4.22)	100
		0.05	188.754 (<0.001)	188.754 (4.55)	100	204.412 (<0.001)	204.412 (4.48)	100	158.097 (<0.001)	158.097 (4.32)	100
0.1	0.45	0.10	102.395 (<0.001)	102.395 (4.63)	100	119.293 (<0.001)	119.293 (4.19)	100	78.159 (<0.001)	78.159 (4.53)	100
		0.50	4.079 (<0.001)	4.079 (4.52)	100	6.534 (<0.001)	6.534 (4.27)	100	3.229 (<0.001)	3.229 (4.32)	100
		1.00	1.259 (<0.001)	1.259 (4.39)	100	1.636 (<0.001)	1.636 (4.28)	100	1.221 (<0.001)	1.221 (4.43)	100
		3.00	1.005 (<0.001)	1.005 (4.27)	100	1.019 (<0.001)	1.019 (4.43)	100	1.006 (<0.001)	1.006 (4.31)	100

,

Table 6. Comparison of the accuracy of the exact formula and NIE technique for the $ARFI(1,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.2,0.25\right)$.

Coefficients		$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.15	0.00	370.026 (<0.001)	370.026 (4.57)	100	370.009 (<0.001)	370.009 (4.26)	100	369.948 (<0.001)	369.948 (4.56)	100
		0.01	327.000 (<0.001)	327.000 (4.26)	100	325.766 (<0.001)	325.766 (4.57)	100	259.746 (<0.001)	259.746 (4.37)	100
		0.05	204.676 (<0.001)	204.676 (4.17)	100	202.294 (<0.001)	202.294 (4.27)	100	97.433 (<0.001)	97.433 (4.73)	100
		0.10	120.055 (<0.001)	120.055 (4.17)	100	118.762 (<0.001)	118.762 (4.47)	100	42.540 (<0.001)	42.540 (4.66)	100
		0.50	6.968 (<0.001)	6.968 (4.68)	100	7.648 (<0.001)	7.648 (4.34)	100	2.322 (<0.001)	2.322 (4.50)	100
		1.00	1.734 (<0.001)	1.734 (4.66)	100	1.944 (<0.001)	1.944 (4.46)	100	1.166 (<0.001)	1.166 (4.49)	100
		3.00	1.025 (<0.001)	1.025 (4.38)	100	1.042 (<0.001)	1.042 (4.44)	100	1.007 (<0.001)	1.007 (4.40)	100
	0.30	0.00	370.090 (<0.001)	370.090 (4.76)	100	369.853 (<0.001)	369.853 (4.47)	100	370.001 (<0.001)	370.001 (4.62)	100
		0.01	326.041 (<0.001)	326.041 (4.55)	100	323.436 (<0.001)	323.436 (4.27)	100	247.444 (<0.001)	247.444 (4.36)	100
		0.05	201.770 (<0.001)	201.770 (4.09)	100	196.389 (<0.001)	196.389 (4.30)	100	86.341 (<0.001)	86.341 (4.71)	100
		0.10	116.911 (<0.001)	116.911 (4.62)	100	112.913 (<0.001)	112.913 (4.56)	100	36.683 (<0.001)	36.683 (4.72)	100
0.1	0.30	0.50	6.460 (<0.001)	6.460 (4.33)	100	6.855 (<0.001)	6.855 (4.26)	100	2.082 (<0.001)	2.082 (4.66)	100
		1.00	1.645 (<0.001)	1.645 (4.38)	100	1.801 (<0.001)	1.801 (4.67)	100	1.133 (<0.001)	1.133 (4.56)	100
		3.00	1.021 (<0.001)	1.021 (4.47)	100	1.034 (<0.001)	1.034 (4.39)	100	1.005 (<0.001)	1.005 (4.55)	100
	0.45	0.00	369.956 (<0.001)	369.956 (4.55)	100	370.231 (<0.001)	370.231 (4.39)	100	370.138 (<0.001)	370.138 (4.52)	100
		0.01	325.256 (<0.001)	325.256 (4.50)	100	322.322 (<0.001)	322.322 (4.63)	100	240.028 (<0.001)	240.028 (4.69)	100
		0.05	199.788 (<0.001)	199.788 (4.59)	100	192.868 (<0.001)	192.868 (4.62)	100	80.310 (<0.001)	80.310 (4.67)	100
		0.10	114.837 (<0.001)	114.837 (4.68)	100	109.416 (<0.001)	109.416 (4.27)	100	33.615 (<0.001)	33.615 (4.64)	100
		0.50	6.147 (<0.001)	6.147 (4.65)	100	6.404 (<0.001)	6.404 (4.36)	100	1.959 (<0.001)	1.959 (4.72)	100
		1.00	1.592 (<0.001)	1.592 (4.76)	100	1.722 (<0.001)	1.722 (4.77)	100	1.117 (<0.001)	1.117 (4.75)	100
		3.00	1.018 (<0.001)	1.018 (4.53)	100	1.030 (<0.001)	1.030 (4.30)	100	1.005 (<0.001)	1.005 (4.86)	100

,

Table 7. Comparison of the accuracy of the exact formula and NIE technique for the $ARFI(2,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.05,0.1\right)$.

Coefficients			$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.2	0.15	0.00	369.988 (<0.001)	369.988 (4.53)	100	370.004 (<0.001)	370.004 (4.54)	100	369.988 (<0.001)	369.988 (4.53)	100
			0.01	322.360 (<0.001)	322.360 (4.48)	100	327.878 (<0.001)	327.878 (4.50)	100	310.417 (<0.001)	310.417 (4.41)	100
			0.05	190.682 (<0.001)	190.682 (4.44)	100	206.917 (<0.001)	206.917 (4.62)	100	164.384 (<0.001)	164.384 (4.44)	100
			0.10	104.406 (<0.001)	104.406 (4.37)	100	122.057 (<0.001)	122.057 (4.57)	100	83.095 (<0.001)	83.095 (4.60)	100
			0.50	4.308 (<0.001)	4.308 (4.61)	100	6.989 (<0.001)	6.989 (4.59)	100	3.526 (<0.001)	3.526 (4.60)	100
0.1	0.2	0.15	1.00	1.289 (<0.001)	1.289 (4.68)	100	1.713 (<0.001)	1.713 (4.45)	100	1.258 (<0.001)	1.258 (4.49)	100
			3.00	1.006 (<0.001)	1.006 (4.60)	100	1.023 (<0.001)	1.023 (4.86)	100	1.007 (<0.001)	1.007 (4.67)	100
		0.30	0.00	370.009 (<0.001)	370.009 (4.45)	100	369.973 (<0.001)	369.973 (4.53)	100	370.183 (<0.001)	370.183 (4.53)	100
			0.01	321.570 (<0.001)	321.570 (4.58)	100	326.862 (<0.001)	326.862 (4.72)	100	306.987 (<0.001)	306.987 (4.57)	100
			0.05	188.414 (<0.001)	188.414 (4.60)	100	203.967 (<0.001)	203.967 (4.47)	100	157.202 (<0.001)	157.202 (4.63)	100
			0.10	102.058 (<0.001)	102.058 (4.52)	100	118.829 (<0.001)	118.829 (4.57)	100	77.442 (<0.001)	77.442 (4.61)	100
			0.50	4.043 (<0.001)	4.043 (4.67)	100	6.463 (<0.001)	6.463 (4.65)	100	3.186 (<0.001)	3.186 (4.60)	100
			1.00	1.255 (<0.001)	1.255 (4.78)	100	1.624 (<0.001)	1.624 (4.38)	100	1.216 (<0.001)	1.216 (4.45)	100
			3.00	1.005 (<0.001)	1.005 (4.73)	100	1.019 (<0.001)	1.019 (4.39)	100	1.006 (<0.001)	1.006 (4.80)	100
		0.45	0.00	370.021 (<0.001)	370.021 (4.55)	100	370.059 (<0.001)	370.059 (4.40)	100	370.009 (<0.001)	370.009 (4.37)	100
			0.01	321.048 (<0.001)	321.048 (4.51)	100	326.287 (<0.001)	326.287 (4.63)	100	304.534 (<0.001)	304.534 (4.49)	100
			0.05	186.930 (<0.001)	186.930 (4.58)	100	202.102 (<0.001)	202.102 (4.45)	100	152.673 (<0.001)	152.673 (4.54)	100
			0.10	100.537 (<0.001)	100.537 (4.45)	100	116.785 (<0.001)	116.785 (4.54)	100	74.029 (<0.001)	74.029 (4.45)	100
			0.50	3.879 (<0.001)	3.879 (4.61)	100	6.145 (<0.001)	6.145 (4.55)	100	2.996 (<0.001)	2.996 (4.61)	100
			1.00	1.235 (<0.001)	1.235 (4.39)	100	1.572 (<0.001)	1.572 (4.72)	100	1.193 (<0.001)	1.193 (4.63)	100
			3.00	1.004 (<0.001)	1.004 (4.62)	100	1.017 (<0.001)	1.017 (4.63)	100	1.005 (<0.001)	1.005 (4.58)	100

and

Table 8. Comparison of the accuracy of the exact formula and NIE technique for the $ARFI(2,d)$ process with $\left({\lambda }_1,{\lambda }_2\right)=\left(0.2,0.25\right)$.

Coefficients			$\mathit{\delta }$	${\mathit{k}}_1,{\mathit{k}}_2=0.1,0.2$			${\mathit{k}}_1,{\mathit{k}}_2=0.5,1$			${\mathit{k}}_1,{\mathit{k}}_2=2,5$
${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	d		Exact	NIE	%ACC	Exact	NIE	%ACC	Exact	NIE	%ACC
0.1	0.2	0.15	0.00	369.951 (<0.001)	369.951 (4.50)	100	369.966 (<0.001)	369.966 (4.35)	100	370.106 (<0.001)	370.106 (4.47)	100
			0.01	326.124 (<0.001)	326.124 (4.40)	100	323.970 (<0.001)	323.970 (4.37)	100	249.873 (<0.001)	249.873 (4.50)	100
			0.05	202.284 (<0.001)	202.284 (4.63)	100	197.588 (<0.001)	197.588 (4.59)	100	88.372 (<0.001)	88.372 (4.60)	100
			0.10	117.497 (<0.001)	117.497 (4.20)	100	114.069 (<0.001)	114.069 (4.41)	100	37.732 (<0.001)	37.732 (4.44)	100
			0.50	6.556 (<0.001)	6.556 (4.49)	100	7.004 (<0.001)	7.004 (4.42)	100	2.124 (<0.001)	2.124 (4.44)	100
			1.00	1.662 (<0.001)	1.662 (4.49)	100	1.828 (<0.001)	1.828 (4.29)	100	1.139 (<0.001)	1.139 (4.45)	100
			3.00	1.021 (<0.001)	1.021 (4.72)	100	1.035 (<0.001)	1.035 (4.70)	100	1.006 (<0.001)	1.006 (4.65)	100
		0.30	0.00	69.915 (<0.001)	69.915 (4.40)	100	370.008 (<0.001)	370.008 (4.53)	100	369.968 (<0.001)	369.968 (4.55)	100
			0.01	325.081 (<0.001)	325.081 (4.45)	100	321.834 (<0.001)	321.834 (4.31)	100	238.438 (<0.001)	238.438 (4.45)	100
			0.05	199.367 (<0.001)	199.367 (4.58)	100	192.008 (<0.001)	192.008 (4.46)	100	79.133 (<0.001)	79.133 (4.57)	100
			0.10	114.403 (<0.001)	114.403 (4.51)	100	108.635 (<0.001)	108.635 (4.53)	100	33.028 (<0.001)	33.028 (4.59)	100
			0.50	6.084 (<0.001)	6.084 (4.40)	100	6.313 (<0.001)	6.313 (4.63)	100	1.936 (<0.001)	1.936 (4.57)	100
			1.00	1.581 (<0.001)	1.581 (4.51)	100	1.707 (<0.001)	1.707 (4.67)	100	1.114 (<0.001)	1.114 (4.56)	100
			3.00	1.018 (<0.001)	1.018 (4.56)	100	1.029 (<0.001)	1.029 (4.53)	100	1.004 (<0.001)	1.004 (4.60)	100
		0.45	0.00	370.028 (<0.001)	370.028 (4.48)	100	369.961 (<0.001)	369.961 (4.43)	100	370.000 (<0.001)	370.000 (4.47)	100
			0.01	324.514 (<0.001)	324.514 (4.45)	100	320.380 (<0.001)	320.380 (4.50)	100	231.538 (<0.001)	231.538 (4.47)	100
			0.05	197.539 (<0.001)	197.539 (4.45)	100	188.440 (<0.001)	188.440 (4.52)	100	74.031 (<0.001)	74.031 (4.49)	100
0.1	0.2	0.45	0.10	112.450 (<0.001)	112.450 (4.55)	100	105.259 (<0.001)	105.259 (4.55)	100	30.507 (<0.001)	30.507 (4.47)	100
			0.50	5.797 (<0.001)	5.797 (4.60)	100	5.913 (<0.001)	5.913 (4.66)	100	1.837 (<0.001)	1.837 (4.59)	100
			1.00	1.534 (<0.001)	1.534 (4.54)	100	1.638 (<0.001)	1.638 (4.58)	100	1.100 (<0.001)	1.100 (4.56)	100
			3.00	1.016 (<0.001)	1.016 (4.59)	100	1.025 (<0.001)	1.025 (4.62)	100	1.004 (<0.001)	1.004 (4.60)	100

, which demonstrate that the Exact technique and the NIE method provide identical ARL values with 100% accuracy. The Exact technique’s main benefit is its computing efficiency; it constantly executes in less than 0.001 s, whereas the NIE method takes several seconds. Additionally, the Exact method is insensitive to variations in $k_1,k_2$ and stable across a variety of long-range dependence levels (d = 0.15, 0.30, 0.45). These findings demonstrate its usefulness as a preferred tool for tracking processes with autoregressive behavior and fractional integration, as well as its robustness and dependability.

Table 9. Comparison between Explicit Formulas and NIE Method.

Process Type	Method	ARL Accuracy	CPU Time (s)	Notes
ARI	Explicit	Identical to NIE	<0.001	Fast, closed-form solution
	NIE	Identical to Explicit	3–5 (depends on number of discretization points)	Requires iterative numerical integration
ARFI	Explicit	Identical to NIE	<0.001	Robust under long memory
	NIE	Identical to Explicit	3–5 (depends on number of discretization points)	Higher computational cost

shown both methods yield the same ARL values; explicit formulas provide overwhelming computational efficiency. In NIE, CPU time increases with the number of discretization points used in the approximation.

4.2. Compare the Between Several Control Charts

The performance of the EWMA and the MEWMA control charts under different situations is compared with that of DMEWMA control charts under ARI(p,d) and ARFI(p,d) models utilizing explicit ARL calculation formulae. To assess how well the control charts identify out-of-control situations, an overall performance indicator, namely the relative mean index, is applied in addition to ARL. Control schemes with the lowest RMI values is considered more effective, as they detect shifts more swiftly and robustly.

$$RMI={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\frac{AR{L}_1\left(r_i\right)-AR{L}_1^{*}\left(r_i\right)}{AR{L}_1^{*}\left(r_i\right)}},$$

where $n$ is number of shifts size considered, $AR{L}_1\left(r_i\right)$ is the $AR{L}_1$ values of a control charts for the specific shift $r_i$ and $AR{L}_1^{*}\left(r_i\right)$ is the smallest of $AR{L}_1$ values found in all of the charts proposed to detect the shift $r_i$, receptively.

Table 10. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart on the $ARI(p,1)$ process with ${\lambda }_1=0.2, {\lambda }_2=0.25,$ and $k_1=0.5, k_2=1$ for $AR{L}_0=370$.

$\mathit{\delta }$	$\mathit{A}\mathit{R}\mathit{I}(1,1)$$: {\mathit{\varphi }}_1=0.1$			$\mathit{A}\mathit{R}\mathit{I}(2,1)$$: {\mathit{\varphi }}_1=0.1,{\mathit{\varphi }}_2=0.2$
	EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
0.00	369.956	369.964	370.097	369.983	370.166	369.885
0.01	301.004	97.7466	78.975	298.315	93.114	75.763
0.05	157.635	24.1823	19.288	152.972	22.7065	18.317
0.1	86.968	12.2099	10.092	83.2523	11.4257	9.554
0.5	7.90502	2.56607	2.545	7.3212	2.41289	2.407
1	2.50098	1.59781	1.678	2.33812	1.52735	1.602
3	1.1337	1.131	1.183	1.11426	1.11131	1.156
RMI	3.379	0.118	0.016	3.430	0.111	0.015

Bold represents the lowest values.

and

Table 11. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart on the $ARI(p,1)$ process with ${\lambda }_1=0.2, {\lambda }_2=0.25,$ and $k_1=0.5, k_2=1$ for $AR{L}_0=500$.

$\mathit{\delta }$	$\mathit{A}\mathit{R}\mathit{I}(1,1)$$: {\mathit{\varphi }}_1=0.1$			$\mathit{A}\mathit{R}\mathit{I}(2,1)$$: {\mathit{\varphi }}_1=0.1,{\mathit{\varphi }}_2=0.2$
	EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
0.00	499.941	499.951	500.131000	499.856	500.014	500.972000
0.01	436.762	232.09	206.723	368.324	204.791	188.945
0.05	213.02	78.679	59.065	227.945	35.059	28.548
0.1	167.524	24.5	17.638	147.68	19.026	13.527
0.5	11.682	5.468	4.439	22.261	5.642	5.484
1	3.38	2.159	2.268	7.992	1.738	2.077
3	1.532	1.528	1.599	1.804	1.246	1.223
RMI	2.403	0.179	0.016	4.164	0.128	0.033

Bold represents the lowest values.

chow the consistent outcomes are seen by comparing the EWMA, MEWMA, and DMEWMA charts using the ARI(p,1) framework for both settings with $AR{L}_0=370$ and $AR{L}_0=500.$ All charts keep the in-control ARL near the objective when there is no shift $\left(\delta =0\right).$ However, the DMEWMA chart shows the lowest out-of-control ARL values across both $ARI(1,1)$ and $ARI(2,1)$ after modest to moderate shifts occur $\left(\delta =0.01-0.1\right),$ showing improved sensitivity. Although the performance difference narrows, DMEWMA still produces lower ARL values than EWMA and MEWMA when the shift size rises $\left(\delta \ge 0.5\right).$ This advantage is further confirmed by the RMI: DMEWMA regularly achieves the lowest RMI (0.016 for $AR{L}_0=370$ and 0.033 for $AR{L}_0=500$), whereas EWMA produces the highest values, indicating a significantly lower detecting capability. The MEWMA performs reasonably well, outperforming EWMA but coming short of DMEWMA. Together, these results demonstrate that DMEWMA provides more efficient and robust detection of mean shifts in autoregressive integrated processes, regardless of the choice of ARL₀.

Table 12. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart on the $ARFI(p,0.45)$ process with $k_1=2, k_2=5.$ for $AR{L}_0=370$.

${\mathit{\lambda }}_1,{\mathit{\lambda }}_2$	$\mathit{\delta }$	$\mathit{A}\mathit{R}\mathit{F}\mathit{I}(1,0.45)$$: {\mathit{\varphi }}_1=0.1$			$\mathit{A}\mathit{R}\mathit{F}\mathit{I}(2,0.45)$$: {\mathit{\varphi }}_1=0.1,{\mathit{\varphi }}_2=0.2$
		EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
0.05, 0.1	0.00	369.440	369.945	369.944	369.896	370.015	370.009
	0.01	297.653	59.074	307.294	297.433	57.928	304.534
	0.05	130.807	14.050	158.097	129.736	13.684	152.673
	0.10	51.174	7.503	78.159	50.331	7.275	74.029
	0.50	1.233	2.166	3.229	1.218	2.081	2.996
	1	1.005	1.535	1.221	1.005	1.482	1.193
	3	1.000	1.156	1.006	1.000	1.134	1.005
	RMI	3.028	0.240	4.285	4.632	0.093	2.048
0.2, 0.25	0.00	369.786	370.075	370.138	370.040	369.983	370.000
	0.01	288.268	55.129	240.028	285.035	53.3156	231.538
	0.05	139.867	13.066	80.310	135.017	12.551	74.031
	0.10	75.271	7.017	33.615	71.676	6.721	30.507
	0.50	7.341	2.091	1.959	6.784	1.999	1.837
	1	2.496	1.504	1.117	2.330	1.441	1.100
	3	1.153	1.149	1.005	1.130	1.125	1.004
	RMI	3.089	0.220	4.207	4.617	0.086	1.963

Bold represents the lowest values.

and

Table 13. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart on the $ARFI(p,0.45)$ process with $k_1=2, k_2=5.$ for $AR{L}_0=500$.

${\mathit{\lambda }}_1,{\mathit{\lambda }}_2$	$\mathit{\delta }$	$\mathit{A}\mathit{R}\mathit{F}\mathit{I}(1,0.45)$$: {\mathit{\varphi }}_1=0.1$			$\mathit{A}\mathit{R}\mathit{F}\mathit{I}(2,0.45)$$: {\mathit{\varphi }}_1=0.1,{\mathit{\varphi }}_2=0.2$
		EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
0.05, 0.1	0.00	499.243	499.926	499.924	499.859	500.02	500.012
	0.01	302.234	179.83	315.262	424.936	178.281	411.532
	0.05	196.766	48.986	215.645	175.319	48.492	206.315
	0.10	81.154	10.139	135.620	68.015	9.831	100.039
	0.50	2.666	2.927	4.364	2.646	2.812	4.049
	1	1.358	2.074	1.650	1.358	2.003	1.612
	3	1.351	1.562	1.359	1.351	1.532	1.358
	RMI	1.784	0.130	2.898	1.653	0.112	2.410
0.2, 0.25	0.00	499.711	500.101	500.186	500.054	499.977	500.107
	0.01	389.551	174.499	299.362	385.182	172.048	298.889
	0.05	189.009	47.657	108.527	182.455	46.961	100.042
	0.10	101.718	9.482	85.426	96.859	9.082	81.226
	0.50	9.92	2.826	2.647	9.168	2.701	2.482
	1	3.373	2.032	1.509	3.149	1.947	1.486
	3	1.558	1.553	1.358	1.527	1.52	1.357
	RMI	3.009	0.093	1.667	2.955	0.086	1.635

Bold represents the lowest values.

present the performance comparison of DMEWMA, MEWMA, and EWMA charts under the $ARFI(p,0.45)$ process with $k_1=2, k_2=5,$$AR{L}_0=370.$ and $AR{L}_0=500,$ respectively. Across both tables, the results show that the MEWMA chart generally outperforms the EWMA and the DMEWMA in detecting small shifts $\left(\delta =0.01-0.1\right)$, as reflected by its consistently lower out-of-control ARL values. This is particularly evident when ${\lambda }_1=0.05, {\lambda }_2=0.1$, where the MEWMA achieves the smallest ARL values across both $ARFI(1,0.45)$ and $ARFI(2,0.45).$ For moderate and larger shifts $\left(\delta \ge 0.50\right)$, all three charts tend to converge in performance, although DMEWMA occasionally achieves slightly lower ARL values (e.g., at $\delta =0.50$, and $\delta =1$). The RMI further highlights this distinction: in both Table 12 and Table 13, MEWMA consistently yields the lowest RMI values (e.g., 0.240 and 0.220 in Table 12, and 0.130 and 0.093 in Table 13), confirming its superior overall detection efficiency for long-memory processes. By contrast, DMEWMA exhibits higher RMI values, reflecting weaker relative performance in this setting, while EWMA performs worst overall with the largest RMI.

5. Real Data Applications

To complement the findings from the simulation study, we further assess the practical relevance of the proposed control chart formulations using real-world datasets. The purpose of this section is to demonstrate how the theoretical results translate into empirical performance when applied to time series exhibiting long-memory and fractional integration characteristics. Two datasets are analyzed: (i) per capita plastic waste generation (kg/person/day) across countries, and (ii) the percentage of government expenditure. Each dataset is modeled within the ARI(p,d) and ARFI(p,d) framework, and the modified EWMA-type charts—including the EWMA, the MEWMA, and the DMEWMA—are applied for monitoring shifts in the underlying processes. The series are decomposed into fitted values and residuals, with model parameters estimated using EViews 10. Finally, the residuals are evaluated for exponential white noise behavior using the Kolmogorov–Smirnov test in SPSS 20, ensuring the suitability of the assumed error structure.

Application 1. 

Per capita plastic waste (kg/person/day) data in 2010 for all countries in the world, with a total of 186 observations. These data were sourced from Our World in Data (https://ourworldindata.org/grapher/plastic-waste-per-capita?tab=table) accessed on 26 May 2025. The Kolmogorov–Smirnov test shows that the residuals of the $ARI\left(2,2\right)$ model follow an exponential distribution (Sig = 0.848 > 0.05). The following equation is suitable:

$$X_t=-0.001+{\epsilon }_t-0.950X_{t-1}-0.524X_{t-2}+\dots ;{\epsilon }_t\sim Exp\left(0.1397\right)$$

(17)

Application 2. 

The government expenditure on education, total (% of government expenditure) from 76 countries worldwide in 2023. These data were sourced from Our World in Data (https://ourworldindata.org/data?topics=Education+and+Knowledge) and accessed on 26 May 2025. The Kolmogorov–Smirnov test shows that the residuals of the $ARFI\left(1,0.45\right)$ model follow an exponential distribution (Sig = 0.147 > 0.05). The following equation is suitable:

$$X_t=-0.048+{\epsilon }_t-0.445X_{t-1}+\dots ;{\epsilon }_t\sim Exp\left(4.7117\right)$$

(18)

Figure 1 shows the detailed context and interpretation for both datasets as follows:

Per capita plastic waste generation (kg/person/day, 2010): This dataset includes 186 observations for all countries worldwide, sourced from Our World in Data. The data represent daily plastic waste generated per person in each country. The time series was modeled using the $ARI\left(2,2\right)$ framework, and the residuals were tested for exponential white noise using the Kolmogorov–Smirnov test (Sig = 0.848 > 0.05), confirming the suitability of the assumed error distribution. Figure 1a shows the observed values alongside the fitted model, illustrating variability across countries and allowing for monitoring shifts in the process using the proposed EWMA-type charts.

Government expenditure on education (% of total expenditure, 2023): This dataset includes 76 countries worldwide, sourced from Our World in Data. The series represents the proportion of government expenditure allocated to education in each country. An $ARFI\left(1,0.45\right)$ model was applied, with residuals tested for exponential white noise (Sig = 0.147 > 0.05), confirming the appropriateness of the model assumptions. Figure 1b shows the observed series and fitted model, highlighting cross-country variability and supporting the evaluation of control chart performance for detecting process shifts.

Table 14. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart for the $ARI(2,2)$ process with ${\lambda }_1=0.2, {\lambda }_2=0.25$ on per capita plastic waste data.

$\mathit{\delta }$	${\mathit{k}}_1=0.1,{\mathit{k}}_2=0.2$			${\mathit{k}}_1=0.5,{\mathit{k}}_2=1$
	EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
	0.75999	0.003474	0.001192	0.75999	0.13154	0.06252
0.00	369.954	370.001	369.934	369.954	369.979	369.932
0.01	282.455	203.047	183.890	282.455	236.623	222.684
0.05	164.298	34.173	23.392	164.298	62.093	48.874
0.1	72.886	8.711	5.138	72.886	21.405	14.797
0.5	21.286	1.158	1.049	21.286	2.034	1.477
1	3.148	1.041	1.011	3.148	1.345	1.143
3	2.045	1.002	1.008	2.045	1.080	1.031
RMI	7.032	0.232	0.001	3.784	0.230	0.000

Bold represents the lowest values.

and

Table 15. Comparison of the performance of the DMEWMA, MEWMA and EWMA control chart for the $ARFI\left(1,0.45\right)$ process with government expenditure on education data.

${\mathit{\lambda }}_1,{\mathit{\lambda }}_2$	$\mathit{\delta }$	${\mathit{k}}_1=0.1,{\mathit{k}}_2=0.2$			${\mathit{k}}_1=0.5,{\mathit{k}}_2=1$
		EWMA	MEWMA	DMEWMA	EWMA	MEWMA	DMEWMA
		0.0241	0.76735	0.139	0.0241	3.03214	0.248686
0.05, 0.1	0.00	370.041	370.01	369.975	370.041	369.922	370.284
	0.01	362.915	359.058	341.737	362.915	326.058	243.361
	0.05	336.021	320.024	260.423	336.021	218.252	102.868
	0.10	305.718	279.760	198.994	305.718	151.214	59.930
	0.50	152.867	121.193	61.709	152.867	34.827	14.224
	1	73.411	58.505	29.270	73.411	4.827	7.5839
	3	10.286	11.284	7.304	10.286	2.819	3.096
	RMI	0.714	0.532	0.000	5.577	0.739	0.112
0.2, 0.25	$\delta$	1.58334	0.97052	0.17921	0.97052	4.05968	0.269323
	0.00	369.887	369.901	370.095	369.901	369.986	370.064
	0.01	339.266	307.101	277.099	307.101	282.765	233.025
	0.05	252.353	182.062	138.064	182.062	143.169	94.282
	0.10	188.439	119.858	84.650	119.858	86.576	54.326
	0.50	52.142	30.1997	20.342	30.1997	17.221	12.948
	1	22.447	14.7324	10.360	14.7324	7.253	7.0162
	3	4.633	4.54435	3.704	4.54435	1.968	2.984
	RMI	0.877	0.329	0.000	1.033	0.282	0.086

Bold represents the lowest values.

present the performance of the EWMA, the MEWMA, and the DMEWMA charts applied to the plastic waste and government expenditure datasets, respectively. Across both applications, the DMEWMA consistently achieves lower ARL values for small and moderate shifts, indicating superior sensitivity compared with the conventional EWMA and the MEWMA procedures. For larger shifts, the performance of all three methods converges, yet the RMI values clearly confirm the efficiency of DMEWMA. These findings highlight the robustness of DMEWMA in real-world contexts, offering more effective detection of subtle changes in both environmental and economic processes.

6. Conclusions

The findings from both the simulation studies and the empirical applications offer a consistent perspective on the relative performance of the three control charts examined: EWMA, MEWMA, and DMEWMA. Overall, although the EWMA chart is simple and widely adopted, it responds more slowly to small process shifts. This behavior aligns with its design principle as a smoothing chart that is better suited for gradual rather than abrupt changes. The MEWMA chart, by incorporating additional memory through a multivariate structure, demonstrates improved sensitivity in certain settings, particularly for moderate shifts. However, the DMEWMA chart consistently emerges as the most efficient alternative, particularly when early detection of subtle changes is required. Its double modification enables a sharper balance between variance reduction and shift detection, resulting in lower ARL values across a broad range of scenarios.

When comparing the simulation results with the empirical applications, a clear consistency is observed. In the simulation framework, which examined the ARI(p,d) and ARFI(p,d) processes, DMEWMA repeatedly outperformed the other methods for small to moderate shifts, whereas all three approaches converged in performance for larger shifts. The real data applications, involving per capita plastic waste and government expenditure, confirmed this pattern. DMEWMA again provided the lowest ARL values and the most favorable RMI, demonstrating its robustness in practical monitoring tasks.

Importantly, the findings suggest that the theoretical advantages established under controlled simulations are preserved in real-world contexts. This reinforces the suitability of DMEWMA as a reliable monitoring tool for processes with long-memory and fractional integration characteristics. At the same time, the comparative analysis indicates that while MEWMA can occasionally compete with DMEWMA under specific parameterizations, EWMA consistently lags behind in detecting small deviations.

Although the DMEWMA chart generally outperforms traditional MEWMA and EWMA charts in detecting moderate to large shifts, the results reveal that under certain long-memory ARFI conditions, the MEWMA chart can exhibit slightly superior performance. This behavior can be explained by the intrinsic interaction between the chart’s weighting mechanism and the long-range dependence of the process.

For ARFI processes, the fractional differencing parameter introduces persistent autocorrelation over long horizons. In such cases, the dual-memory structure of DMEWMA—where two smoothing parameters (λ₁ and λ₂) are combined—may unintentionally over-smooth or overreact to correlated residuals. Specifically, when both λ₁ and λ₂ are relatively small, the chart gives high weight to past observations, amplifying the long-memory effects already present in the data. As a result, the control statistic becomes less responsive to small deviations, leading to slightly longer out-of-control run lengths.

In contrast, the standard MEWMA chart, which relies on a single smoothing parameter λ, maintains a more stable balance between sensitivity and robustness in long-memory environments. Its simpler exponential weighting may align better with the decay rate of autocorrelation in ARFI processes, allowing it to track persistent but gradual mean shifts more efficiently.

Therefore, the slight inferiority of DMEWMA in these cases is not due to a structural weakness, but rather to the complex interaction between dual-memory smoothing and long-range dependence. Fine-tuning the parameter pair (λ₁, λ₂) for ARFI-specific memory characteristics could potentially mitigate this effect, and future studies could explore adaptive tuning strategies that dynamically adjust the DMEWMA parameters based on the estimated fractional differencing parameter (d).

The consistency between simulation and empirical evidence highlights the broader implication: control charts designed for ARFI-type processes are not merely theoretical constructs but can be effectively applied to complex real data exhibiting persistence and structural dependencies. This echoes earlier work emphasizing the role of long-memory modeling in economic and environmental data [10,11]. Overall, this study provides strong evidence that DMEWMA offers a significant advancement over conventional EWMA-type charts for both simulated and real-world monitoring applications.

Although this study focuses on the statistical evaluation of the DMEWMA chart, future work may incorporate an economic design framework. The DMEWMA parameters can be optimized to minimize the expected total cost per unit time, accounting for sampling, false alarm, and process adjustment costs. Integrating the proposed chart into an economic optimization model would further enhance its industrial applicability.