Distribution-Free EWMA Scheme for Joint Monitoring of Location and Scale Based on Post-Sales Online Review Process

Abstract

Nowadays, the online comment process of product after-sales has become a key part of product development. Quality problems, such as the failure of products or services, are more likely to exist or hide in negative comments. Therefore, this paper focuses on detecting abnormal changes in both the time between review T and the emotional score S of negative comments. Due to the complexity of the online review process, the distribution assumption of S and T may be invalid. To solve this problem, this study propose a distribution-free monitoring scheme that combines the exponentially weighted moving average-based Lepage statistics of S and T using a max-type combining function. This scheme is designed for joint monitoring of location and scale parameters in Phase II of an unknown but continuous process. The scheme’s performance is evaluated via Monte Carlo simulation under in-control and out-of-control conditions, using statistical measures such as the mean, standard deviation, median, and selected percentiles of the run length distribution. Simulation results indicate that the scheme is effective in detecting shifts in both location and scale parameters. Furthermore, an application of the proposed scheme for monitoring online reviews is discussed to illustrate its implementation design.

1. Introduction

Driven by the rapid development of information technology in the e-commerce era, an increasing number of customers tend to publish their product and service reviews online. Customer satisfaction is determined by the difference between customers’ initial expectation and perceived quality of products or services (see Elkhani et al. [1], Andrejić et al. [2], and Song et al. [3]). These online reviews generate online reputations and affect future customers’ decision to buy products (see Book et al. [4]). This phenomenon has attracted more and more attention in recent studies, and online customer reviews embody rich information. Usually, online customer ratings are used as an indicator of overall customer satisfaction in various studies (see De et al. [5] and Yang et al. [6]). To apply customer satisfaction evaluation to after-sales research, we analyzed user-generated content (UGC).

It is widely recognized that 92% of global consumers trust UGC and peer recommendations [7]. Online comments, a key component of UGC, can be categorized into three sentiment types: negative, neutral, and positive [8]. Recent findings indicate that customers increasingly share their immediate and authentic feedback on online platforms after experiencing products or services. User ratings significantly influence consumer decision making and sales performance, with negative comments differing notably from non-negative ones in terms of quality and impact. Potential customers often pay particular attention to negative comments before making a purchase, as these can reveal underlying quality issues [9]. Consequently, many companies dedicate substantial resources to addressing negative reviews, as they often highlight problems that could lead to customer complaints or product recalls. In this paper, we focus on the extraction and identification of negative comments for stable products or services, aiming to monitor the online feedback process effectively.

The industrial sector has undergone a period of unprecedented expansion and transformation, characterized by rapid technological advancements, accelerated adoption of automation and digitalization, and a significant increase in production capacity, reflecting an exceptionally swift pace of development in recent decades [10,11]. Statistical process monitoring (SPM) plays an important role in various fields, such as manufacturing, business processing, image processing, and medical monitoring [12]. This study primarily investigates nonparametric statistical process monitoring (NSPM), an approach that has gained increasing attention in contemporary applications and is prevalent across numerous domains, as demonstrated in the works of Li et al. [13] and Song et al. [14]. For a comprehensive overview of the SPM framework, interested readers are referred to the articles and textbooks authored by Qiu [15].

Within the NSPM framework, negative comments can be regarded as adverse events and continuously monitored. In SPM, a two-phase implementation is commonly adopted in practice. Phase I focuses on analyzing historical data to establish in-control (IC) process parameters and initial control limits, while Phase II involves ongoing monitoring of the process using these control limits to detect any out-of-control (OOC) signals and ensure sustained stability. However, in Phase II, the number of negative evaluations for general products is often much smaller compared to the number of non-negative evaluations. This makes traditional monitoring schemes less sensitive to changes in the number of negative evaluations [16].

In the past decades, SPM researchers and practitioners have expressed serious concern about the lack of IC robustness of parameter monitoring schemes. In particular, the exponential weighted moving average (EWMA) monitoring scheme has attracted great attention due to its effectiveness in detecting small shifts. For instance, risk management in complex operational systems—such as the transport of dangerous goods—relies on systematic monitoring and early warning mechanisms, where EMEA-based approaches support proactive detection of failures [17]. Similarly, in intelligent transportation systems, robust data processing under adverse conditions (e.g., hazy road images) is essential for reliable monitoring, which parallels the need for robust sentiment signal extraction from noisy user feedback [18]. Moreover, in service-oriented domains like insurance, the handling of customer claims—often triggered by negative experiences—must be monitored and optimized to maintain service quality and trust [19]. Interested readers can refer to articles such as Lucas et al. [20] and Maravelakis and Castagliola [21]. In both parametric and nonparametric cases, the EWMA scheme is more widely used in the monitoring of mean or variance due to its high sensitivity to detect small shifts. For the parametric implementation of the EWMA monitoring scheme, interested readers can refer to Castagliola [22] and Hamilton and Crowder [23]; for the nonparametric version, see Zou et al. [24], Li et al. [25], and Song et al. [26].

Most of the monitoring schemes discussed above are designed to detect shifts in location or scale parameters individually. However, in practice, researchers have increasingly recognized the importance of monitoring both parameters simultaneously [27]. Considerable progress has been made in jointly monitoring the mean and variance of normally distributed processes. However, in the context of distribution-free monitoring schemes, relatively little research has focused on jointly monitoring location and scale parameters. Mukherjee et al. [28] proposed a Shewhart-type Phase-II scheme for this purpose. When both parameters are unknown, the Lepage statistic is used to jointly detect shifts in location and scale. Subsequently, Chowdhury et al. [29] developed a distribution-free cumulative sum (CUSUM) scheme for simultaneously monitoring the location and scale parameters of univariate continuous processes, based on the work of Lepage [30].

In this paper, we propose a distribution-free EWMA scheme based on the Lepage statistic for jointly monitoring location and scale parameters. We denote the time interval between two consecutive negative comments as T, and use the sentiment score S of customer comments as an indicator of product or service quality. An increase in T reflects higher customer satisfaction and implies a reduced need for quality improvement, whereas a decrease in T indicates a greater opportunity for enhancement. For further details, see Wu et al. [31] and Xu [32]. Additionally, for negative comments (where $S<0$), a larger absolute value of S (i.e., more negative) indicates a lower customer evaluation, while a smaller absolute value suggests a more favorable assessment. We assume that S and T are independent. To the best of our knowledge, numerous studies have investigated nonparametric tests for binary processes, including the classical Wilcoxon rank-sum test and the method proposed by Peters and Randles [33].

For online comments, most post-sales evaluations are divided into two types. One is the quantitative numerical rating, which reflects the overall customer evaluation of products or services based on star ratings [34]; the other is qualitative evaluation, which reflects the opinions on or emotions toward products or services according to the customer’s text. Extensive research suggests that comment content is a more effective indicator of customer-perceived quality than numerical ratings. Therefore, we replace numerical scores with emotional scores derived from comment content, assuming that customer satisfaction is determined solely by the perceived quality of the product. In this paper, emotional scores are computed from comment content to monitor the post-sale evaluation process. In emotion analysis, anomaly detection may be necessary, as anomalies can arise from sudden emotional shifts hidden in customer comments. In recent years, there has been growing interest in anomaly detection within affective analysis; for further details, see Rakhshan et al. [35], Soo-Guan et al. [36], and Read and Jonathon [37].

This paper introduces a new SPM tool called the EWMA-Lepage scheme. Its key contribution is a nonparametric method that can simultaneously detect shifts in both the location and scale parameters of a process without assuming a specific data distribution. Specifically, we apply this scheme to monitor customer feedback in an online environment, using it to track two key metrics derived from customer comments: the sentiment score as an indicator of customer evaluation and the time between negative comments. By jointly monitoring these two aspects, the scheme helps businesses quickly identify shifts in customer dissatisfaction, enabling timely quality improvements and ultimately enhancing their online reputation (eWOM). The work fills a gap by extending joint monitoring to a nonparametric, real-world service quality context. The remainder of this paper is organized as follows: Section 2 outlines the design and implementation of the distribution-free EWMA-Lepage scheme for jointly monitoring location and scale parameters. Section 3 investigates the performance of the proposed EWMA-Lepage scheme in both the IC and OOC states. Section 4 presents a real-world example to demonstrate the effectiveness of the proposed scheme. Section 5 concludes the paper by presenting the main findings and identifying opportunities for future research.

2. The EWMA-Lepage Monitoring Scheme

This section will introduce the design and implementation of the distribution-free EWMA-Lepage scheme based on the Lepage statistic. The well-known Lepage statistic is the sum of squares of the standardized Wilcoxon rank-sum (WRS) statistic for the location parameter and the standardized Ansari–Bradley (AB) statistic for the scale parameter. Unlike parametric methods, distribution-free approaches do not require a distinction between cases where process parameters are known and unknown in SPM schemes. However, we need to consider a continuous bivariate process that is not affected by any assignable or special causes, but only by random variation. Such a process is referred to as an IC process in SPM.

In this section, we assume that a reference sample of size m is available from the IC process for the monitoring scheme proposed in this study. At the same time, the samples continuously collected from the real-time online system are referred to as test samples, corresponding to the Phase II data review process. At each stage of the Phase II monitoring, a test sample of size n is collected. The objective of this study is to promptly and accurately identify abnormal shifts in test samples. Therefore, we can transform the monitoring problem into a hypothesis-testing process regarding whether the reference samples and test samples come from the same distribution.

2.1. Lepage Statistic for Bivariate Test

Let $X_0=(X_{01},X_{02},\dots ,X_{0m})$ and $Y_0=(Y_{01},Y_{02},\dots ,Y_{0m})$ denote reference samples of size m, where $X_0$ and $Y_0$ are collected from two unknown continuous distributions $F_0\left(s\right)$ and $G_0\left(t\right)$, respectively. Let $X_1=(X_{11},X_{12},\dots ,X_{1n})$ and $Y_1=(Y_{11},Y_{12},\dots ,Y_{1n})$ denote test samples of size n, where $X_1$ and $Y_1$ are drawn from two unknown continuous distributions $F_1\left(s\right)$ and $G_1\left(t\right)$, respectively. The reference sample $X_0$ and the test sample $X_1$ are combined into a single sample, denoted as S. Similarly, $Y_0$ and $Y_1$ are pooled into another sample, denoted as T. It is assumed that S and T are independent of each other. When $F_0\left(s\right)\ne F_1\left(s\right)$ or $G_0\left(t\right)\ne G_1\left(t\right)$, we can conclude that a shift occurs in either the location parameter, the scale parameter, or both during the Phase II process.

Each newly collected test sample undergoes the following hypothesis testing:

$$H_0:F_0\left(s\right)=F_1\left(s\right)and{G}_0\left(t\right)=G_1\left(t\right),$$

$$H_1:F_0\left(s\right)\ne F_1\left(s\right)or{G}_0\left(t\right)\ne G_1\left(t\right),$$

Accordingly, introduce an indicator variable $I_k=0$ or 1 as the k-th order statistic of T or S among the $N=m+n$ samples. The Lepage statistic is the sum of squares of the the standardized WRS statistic and the standardized AB statistic. WRS statistics are constructed according to the rank sum when the test sample is compared with the reference sample. For S, combining its reference sample with test sample together and arranging all $m+n$ observations in ascending order, the ranks for S should be denoted by $R_S=R_{S,1},R_{S,2},\dots ,R_{S,k},\dots ,R_{S,m+n-1},R_{S,m+n}$. Similarly, the ranks for T can be denoted by $R_T=R_{T,1},R_{T,2},\dots ,R_{T,k},\dots ,R_{T,m+n-1},R_{T,m+n}$. The WRS statistic for the i-th test sample is given by the following expression:

$$T_1=S_1={\sum }_{k=1}^{N}k{I}_k$$

The AB statistic is a nonparametric test for the two-sample scale problem based on the statistic $T_2$, defined as:

$$T_2=S_2={\sum }_{k=1}^N\left|k-{\displaystyle \frac{1}{2}}(N+1)\right|I_k$$

Because extreme values of $T_d$ and/or $S_d$$(d=1,2)$ indicate a departure from the null hypothesis, we propose using the sum of the squares of the WRS and AB statistics. This approach enables us to make decisions based only on the upper control limit. For the i-th test sample, the combination of the standardized WRS and AB statistics forms the Lepage statistic, which is given by:

$$L_{T,i}={\left({\displaystyle \frac{T_1-E\left(T_1\right)}{\sqrt{Var\left(T_1\right)}}}\right)}^2+{\left({\displaystyle \frac{T_2-E\left(T_2\right)}{\sqrt{Var\left(T_2\right)}}}\right)}^2$$

$$L_{S,i}={\left({\displaystyle \frac{S_1-E\left(S_1\right)}{\sqrt{Var\left(S_1\right)}}}\right)}^2+{\left({\displaystyle \frac{S_2-E\left(S_2\right)}{\sqrt{Var\left(S_2\right)}}}\right)}^2$$

The mean and variance of the WRS statistics $S_1$ and $T_1$ are:

$$E\left(T_1\right)=E\left(S_1\right)={\displaystyle \frac{1}{2}}n(N+1)$$

$$Var\left(T_1\right)=Var\left(S_1\right)={\displaystyle \frac{1}{12}}mn(N+1)$$

The mean and variance of the AB statistics $S_2$ and $T_2$ are:

$$E\left(T_2\right)=E\left(S_2\right)=\left\{\begin{array}{cc}{\displaystyle \frac{n\left(N^2-1\right)}{4N}}\hfill & \mathrm{when}N\mathrm{is}\mathrm{odd}\hfill \\ {\displaystyle \frac{nN}{4}}\hfill & \mathrm{when}N\mathrm{is}\mathrm{even},\hfill \end{array}\right.$$

$$Var\left(T_2\right)=Var\left(S_2\right)=\left\{\begin{array}{cc}{\displaystyle \frac{mn(N+1)\left(N^2+3\right)}{48N^2}}\hfill & \mathrm{when}N\mathrm{is}\mathrm{odd}\hfill \\ {\displaystyle \frac{m\left(N^2-4\right)}{48(N-1)}}\hfill & \mathrm{when}N\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

Mukherjee et al. [28] recommend using this statistic for joint monitoring of location and scale parameters of an unknown but continuous process.

2.2. Design of the EWMA-Lepage Scheme

In many industrial settings, rapid detection of shifts in manufacturing processes is crucial. The EWMA schemes are generally more sensitive and effective than Shewhart-type schemes in detecting small to moderate shifts. Therefore, since there is an EWMA scheme based on the Lepage statistic for S and another for T, a straightforward approach is to combine them using a well-known combination function for joint monitoring of S and T. The implementation procedure of the proposed distribution-free EWMA-Lepage scheme is presented in this subsection.

Some distribution-free EWMA schemes based on the Lepage statistic for jointly monitoring location and scale parameters were proposed by Li et al. [13]. We can calculate the values of $L_{S,i}$ and $L_{T,i}$ using the reference sample and the i-th test sample, where $i=1,2,\dots$. The EWMA-based monitoring statistics can be denoted by:

$$Y_{S,i}=r·L_{S,i}+(1-r)·Y_{S,i-1},i=1,2,\dots$$

$$Y_{T,i}=r·L_{T,i}+(1-r)·Y_{T,i-1},i=1,2,\dots$$

where r is a smoothing parameter, and one can easily verify that $E\left(L_{S,i}\right)=E\left(L_{T,i}\right)=1$.

Different from the traditional EWMA method discussed in Li et al. [13], we find that combining two independent EWMA statistics using their maximum value is more appealing than the traditional approach. This study integrates two distinct EWMA statistics—one targeting the location parameter based on the WRS statistic, and the other focusing on the scale parameter derived from the AB statistic. In this paper, by using the max-type combination function, the resulting combined statistic can be denoted as:

$$Y_i=max(Y_{S,i},Y_{T,i}),i=1,2,\dots$$

If $Y_i$ exceeds the control limit h, a signal will be generated. The corresponding flowchart is shown in Figure 1.

2.3. Implementation of the EWMA-Lepage Scheme

The EWMA-Lepage scheme can be implemented in practice by the following steps:

Step 1.

Based on historical process data under IC condition, a reference sample of size m is collected.

Step 2.

The i-th test sample of size n is collected from real-time online review data.

Step 3.

Calculate the statistics $S_{1,i}$, $S_{2,i}$, $T_{1,i}$, and $T_{2,i}$ between the reference sample and the i-th test sample, and obtain their means and variances depending on whether $N=m+n$ is even or odd.

Step 4.

Calculate the Lepage statistics $L_{S,i}$ and $L_{T,i}$ mentioned in Section 2.1 for the i-th test sample. Then, calculate the plooting statistic $Y_i=max(Y_{S,i},Y_{T,i}),i=1,2,\dots$ for the i-th test sample.

Step 5.

Set the control limit h.

Step 6.

If $Y_i$ exceeds h, the process is considered to be OOC at the i-th test sample. Otherwise, the process is considered to be IC, and the next sample is taken.

Step 7.

When an OOC signal is triggered at the i-th test sample, it is concluded that the m reference samples and the n test samples come from different distributions, and the cause of the alarm is subsequently investigated. Therefore, this section adopts the following three rules:

i.

If $Y_{T,i}\le h<Y_{S,i}$, then a shift in S is detected.

ii.

If $Y_{S,i}\le h<Y_{T,i}$, then a shift in T is detected.

iii.

If both $Y_{S,i}$ and $Y_{T,i}$ exceed h, then shifts in both S and T are detected.

2.4. Determination of the Control Limit

The average run length $\left(ARL\right)$ serves as a key performance metric for evaluating SPM schemes, particularly due to its high sensitivity in detecting shifts in the process mean or variance. When the process is IC, the $ARL$ represents the average number of samples collected before a false alarm is triggered and the $ARL$ under the IC and OOC conditions are denoted as $AR{L}_0$ and $AR{L}_1$, respectively. The design objective of this metric is to ensure effective detection of genuine process shifts while maintaining a low rate of false alarms. When the process is OOC, samples affected by special causes should be continuously monitored until an OOC signal is triggered. The control limit h dependes on the following four design parameters: (1) reference sample size m, (2) test sample size n, (3) the value of $AR{L}_0$, (4) the smoothing parameter r. The selection of the parameter r follows the approach proposed by Graham et al. [38] for the distribution-free EWMA scheme. From this study, we conclude that $r=0.05$ is typically used for detecting small shifts, $r=0.1$ is suitable for moderate shifts, and $r=0.2$ performs better for large shifts.

In this paper, we also investigate the effect of the smoothing parameter r on the performance of monitoring schemes by considering five values: $r=0.05,0.1,0.2,0.3,0.5$. It is well recognized that deriving a closed-form expression for control limits in the EWMA scheme is difficult. To address this issue, Li et al. [39] and several other researchers proposed using the Markov chain method to approximate $ARL$, which can then be used to estimate the control limits. However, in the current situation of this paper, due to the distribution being unknown, it is difficult to deduce the appropriate approximation of $ARL$ by Markov chain method. Therefore, we use another method, the Monte Carlo simulation, to calculate $ARL$ and estimate the control limit. This method was proposed by Qiu [15] and has made a great contribution to obtaining the control limit.

To implement the proposed scheme, we determine the control limit based on a specified $AR{L}_0$ target. It is also assumed that, under the IC state, the variables S and T follow normal and exponential distributions, respectively. We employ the Monte Carlo simulation method based on 50,000 replications to determine the values of the control limit. Since the proposed scheme is distribution-free, we generally generate m known reference samples and n unknown test samples from the same distribution. Since the control limit h is invariant to the distributions of S and T, it can be determined solely based on the predetermined parameters $(m,n,r,AR{L}_0)$ in the EWMA-Lepage scheme. We consider reference sample sizes of $m=50,100,150$ and test sample sizes of $n=5,10,15$, with a target $AR{L}_0=370$. The values of the control limit h for different combinations of $(m,n,r)$ under this $AR{L}_0$ setting are presented in

Table 1. Control limits of the EWMA-Lepage scheme for various m and n with $AR{L}_0=370$.

h
m	n	r = 0.5	r = 0.1	r = 0.2
50	5	2.7495	3.3125	4.2773
100	5	2.8068	3.3906	4.4304
150	5	2.8267	3.4208	4.4766
50	10	2.7128	3.2636	4.2217
100	10	2.7969	3.3906	4.4216
150	10	2.8359	3.4291	4.4766
50	15	2.6543	3.2061	4.1484
100	15	2.7734	3.3643	4.3711
150	15	2.8203	3.4200	4.4648

. As shown in Table 1, for fixed values of n and r, the control limit h increases with increasing m. In addition, when m and r are fixed, the control limit h increases as n increases (except for $r=0.05$).

3. Performance Investigation

In this section, we use Monte Carlo simulations to evaluate the IC and OOC performance of the proposed monitoring scheme. The average run length (ARL) and the standard deviation of the run length (SDRL) are commonly used performance metrics in SPM. Furthermore, because the run length is a random variable and its distribution is right-skewed, it is important to examine not only the $ARL$ but also selected percentiles, such as the 5th, 25th, 50th, 75th, and 95th percentiles. First, we examine the IC run length distribution of the proposed monitoring schemes. Then, we evaluate their OOC performance under combinations of four different distributions.

3.1. IC Performance

The nonparametric monitoring scheme exhibits IC robustness; that is, it maintains consistent IC performance across all types of continuous distributions. Therefore, to investigate the IC performance, we set S and T to follow normal and exponential distributions, respectively. As mentioned earlier, we consider reference sample sizes $m=50,100,150$ and test sample sizes $n=5,10,15$. For each combination of $(m,n)$ with a fixed ${\mathrm{ARL}}_0=370$, the corresponding control limits h are presented in Table 1. These values are then used to compute the mean, standard deviation (SD), and selected percentiles (5th, 25th, 50th, 75th, and 95th) of the run length distribution, as summarized in

Table 2. IC performance of the EWMA-Lepage scheme $(AR{L}_0=370)$.

r	m	n	ARL	SDRL	5th	25th	50th	75th	95th
0.05	50	5	366.44	526.00	13	48	137	403	2000
	100	5	369.77	474.87	18	69	182	451	1585
	150	5	369.54	445.38	20	80	202	468	1418
	50	10	372.79	568.44	10	36	113	406	2000
	100	10	368.04	498.74	14	55	157	440	1744
	150	10	373.11	467.83	17	70	189	467	1532
0.1	50	5	371.17	504.71	12	54	159	447	1805
	100	5	370.70	456.60	15	73	194	471	1460
	150	5	369.82	430.61	18	83	213	480	1349
	50	10	369.15	536.65	9	39	132	427	2000
	100	10	369.72	477.12	13	65	181	455	1597
	150	10	369.25	441.62	17	77	202	475	1394
0.2	50	5	367.55	480.80	12	60	171	453	1603
	100	5	373.95	440.85	15	81	210	488	1387
	150	5	370.99	416.61	16	89	226	490	1270
	50	10	371.20	511.30	9	48	155	447	1821
	100	10	372.53	461.72	13	70	195	474	1495
	150	10	367.92	432.88	15	78	208	479	1344

.

It can be observed from Table 2 that, under the IC condition, the actual $ARL$ values are approximately equal to the target ${\mathrm{ARL}}_0=370$. This indicates that the IC performance of the EWMA-Lepage scheme remains highly satisfactory. Furthermore, the run length distribution in the IC state exhibits a strong right skew, as evidenced by the fact that, for all combinations of $(m,n,r)$, the $ARL$ is significantly greater than the median. When n is fixed, as m increases, the values of all percentiles increase (except for the 95th percentile), and SDRL decreases. This indicates that the run length distribution becomes more symmetric and the performance of the scheme improves.

On the contrary, when m is fixed, SDRL increases as n increases, while most percentile values decrease (excluding the 95th percentile). This indicates that the run length distribution gradually becomes more right-skewed. Therefore, it is recommended to use relatively large m and small n, as this combination reduces the value of SDRL and enhances the overall performance of the monitoring scheme. These results are also consistent with other nonparametric schemes, such as Zhang et al. [40] and Huang et al. [41]. In addition, when S and T follow other distributions, the simulation results are expected to be broadly consistent with those presented in Table 2.

3.2. OOC Performance

In this subsection, we evaluate the OOC performance of the proposed EWMA-Lepage scheme using several metrics: ARL, SDRL, and selected percentiles of the run length distribution. These metrics provide meaningful insights into the scheme’s detection capability. Furthermore, the performance of the EWMA-Lepage scheme is evaluated under several different distributions as part of the comparative study. As noted by Zhang et al. [42], it is not feasible to use existing distribution models for a comprehensive comparison of distribution-free monitoring schemes. Therefore, we compare the distribution-free monitoring schemes based on the four distributions proposed by Zhang et al. [40]. We consider the following representative distributions: S follows either a normal distribution (representing symmetry) or a gamma distribution (representing high skewness), while T follows either an exponential distribution (representing thin tails) or a Weibull distribution (representing heavy tails).

3.2.1. Analysis Under Pattern 1: $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Exp\left(\lambda \right)$

In this comparative study, we also consider several values of $r=0.05,0.1,0.2,0.3,0.5$ and set $AR{L}_0=370$. The classification of shift sizes was discussed by Zhang et al. [43]. To better characterize the size of shifts, we introduce two parameters, ${\xi }_1$ to represent the shift in T and ${\xi }_2$ to represent the shift in S, which are defined as follows:

$${\xi }_1=\lambda /{\lambda }_0 {\xi }_2=a·X+b$$

We assume that under the IC state, $S\sim N({\mu }_0,{{\sigma }_0}^2)$ and $T\sim \mathrm{Exp}\left({\lambda }_0\right)$, where ${\mu }_0$, ${\sigma }_0$, and ${\lambda }_0$ are the known parameters estimated from the reference sample. For the test sample, the parameters of S and T are denoted by $\mu$, $\sigma$, and $\lambda$, respectively, which are monitored for potential shifts from their IC values.

According to Huang et al. [41], the magnitude of a shift can be classified as follows: $\left|{\xi }_i\right|\le 0.5$ represents a small shift, $0.5<\left|{\xi }_i\right|<1.5$ represents a moderate shift, and $\left|{\xi }_i\right|\ge 1.5$ defines a large shift, for $i=1,2$. To simulate shifts of various types and directions, we vary the shift parameters according to the distribution of S. When S follows the normal distribution, we set ${\xi }_1=1.0,1.5,2$, $a=1.0,1.25,1.5,2$, and $b=0,0.25,0.5,0.75,1.0,1.5,2.0$. Therefore, when S follows the normal distribution, ${\xi }_1=1$ and $a=1$, $b=0$ indicate that both the location and scale parameters remain unchanged. For brevity, only the results under the setting $m=100$, $n=5$, and $AR{L}_0=370$ are presented in

Table A1. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Exp\left(\lambda \right)$ ($\lambda =1$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	373.27 (483.59) 17, 70, 178, 450, 1633	375.17 (458.82) 17, 76, 198, 473, 1490	375.75 (446.71) 15, 80, 208, 485, 1409	367.87 (424.95) 15, 84, 212, 482, 1322	369.59 (421.95) 15, 85, 217, 491, 1298
	0.25	177.91 (288.91) 10, 31, 76, 191, 692	198.07 (304.26) 10, 35, 89, 224, 756	212.21 (296.75) 9, 41, 105, 253, 783	221.79 (301.45) 9, 43, 116, 270, 795	227.66 (306.56) 8, 46, 122, 280, 818
	0.5	34.37 (54.20) 5, 11, 20, 38, 104	39.71 (66.23) 4, 11, 21, 43, 133	50.81 (83.00) 3, 11, 25, 56, 179	59.03 (97.92) 3, 11, 28, 67, 216	71.08 (109.14) 3, 14, 36, 83, 254
	0.75	10.46 (8.68) 3, 5, 8, 13, 25	10.70 (10.83) 2, 5, 8, 13, 28	12.46 (15.27) 2, 5, 8, 15, 37	13.97 (17.77) 2, 5, 8, 16, 44	18.54 (24.97) 2, 5, 11, 22, 61
	1	5.37 (3.19) 2, 3, 5, 7, 11	4.99 (3.32) 2, 3, 4, 6, 11	(4.16) 1, 3, 4, 6, 13	5.42 (4.84) 1, 2, 4, 7, 14	6.53 (6.72) 1, 2, 4, 8, 19
	1.5	2.46 (1.07) 1, 2, 2, 3, 4	2.24 (1.06) 1, 2, 2, 3, 4	2.09 (1.05) 1, 1, 2, 3, 4	2.09 (1.16) 1, 1, 2, 3, 4	2.11 (1.37) 1, 1, 2, 3, 5
	2	1.60 (0.62) 1, 1, 2, 2, 3	1.46 (0.58) 1, 1, 1, 2, 2	1.35 (0.54) 1, 1, 1, 2, 2	1.32 (0.53) 1, 1, 1, 2, 2	1.28 (0.53) 1, 1, 1, 1, 2
1.25	0	63.61 (91.95) 7, 18, 37, 73, 201	73.01 (110.11) 6, 18, 40, 85, 243	86.34 (116.15) 6, 21, 49, 107, 294	94.78 (124.96) 5, 22, 54, 118, 320	111.30 (136.91) 6, 27, 66, 144, 370
	0.25	41.25 (57.38) 6, 14, 25, 48, 125	48.02 (71.28) 5, 13, 27, 56, 158	57.35 (81.31) 4, 14, 31, 69, 196	62.92 (82.88) 4, 15, 36, 78, 212	77.04 (99.12) 4, 18, 45, 98, 251
	0.5	17.91 (17.04) 4, 8, 13, 22, 48	19.52 (23.23) 3, 7, 13, 23, 57	22.01 (27.55) 3, 7, 14, 27, 70	26.39 (35.77) 2, 7, 16, 32, 82	31.56 (44.66) 2, 8, 19, 39, 100
	0.75	8.88 (6.31) 2, 5, 7, 11, 20	8.95 (7.50) 2, 4, 7, 11, 22	9.48 (9.78) 2, 4, 7, 12, 26	10.40 (11.59) 2, 4, 7, 13,30	12.67 (14.03) 1, 4, 8, 16, 39
	1	5.37 (3.23) 2, 3, 5, 7, 11	5.06 (3.29) 1, 3, 4, 6, 11	5.01 (3.79) 1, 2, 4, 6, 12	5.34 (4.49) 1, 2, 4, 7, 14	6.14 (6.05) 1, 2, 4, 8, 17
	1.5	2.72 (1.30) 1, 2, 2, 3, 5	2.51 (1.25) 1, 2, 2, 3, 5	2.38 (1.32) 1, 1, 2, 3, 5	2.35 (1.40) 1, 1, 2, 3, 5	2.39 (1.66) 1, 1, 2, 3, 6
	2	1.81 (0.74) 1, 1, 2, 2, 3	1.66 (0.71) 1, 1, 2, 2, 3	1.55 (0.69) 1, 1, 1, 2, 3	1.51 (0.70) 1, 1, 1, 2, 3	1.46 (0.72) 1, 1, 1, 2, 3
1.50	0	18.89 (16.61) 4, 9, 14, 24, 48	19.85 (19.01) 3, 8, 14, 25, 55	23.52 (25.09) 3, 8, 16, 30, 70	28.55 (33.39) 3, 8, 18, 36, 88	37.35 (43.18) 3, 10, 24, 48, 117
	0.25	15.98 (13.20) 4, 8, 12, 20, 39	16.96 (16.01) 3, 7, 12, 21, 47	19.44 (20.52) 3, 7, 13, 24, 58	23.17 (26.70) 2, 7, 15, 30, 70	30.09 (34.25) 2, 8, 19, 40, 95
	0.5	11.16 (8.13) 3, 6, 9, 14, 27	11.13 (9.23) 2, 5, 8, 14, 28	12.11 (11.69) 2, 5, 9, 15, 34	13.65 (14.14) 2, 5, 9, 17, 40	17.17 (19.08) 2, 5, 11, 22, 53
	0.75	7.36 (4.68) 2, 4, 6, 9, 16	7.06 (5.02) 2, 4, 6, 9, 17	7.48 (6.33) 2, 3, 6, 10, 19	8.03 (7.40) 1, 3, 6, 10, 22	9.28 (9.32) 1, 3, 6, 12, 27
	1	5.16 (3.01) 2, 3, 5, 7, 11	4.87 (3.09) 1, 3, 4, 6, 11	4.76 (3.43) 1, 2, 4, 6, 11	4.93 (3.90) 1, 2, 4, 6, 12	5.59 (5.03) 1, 2, 4, 7, 15
	1.5	2.93 (1.41) 1, 2, 3, 4, 6	2.71 (1.42) 1, 2, 2, 3, 5	2.58 (1.49) 1, 2, 2, 3, 5	2.55 (1.58) 1, 1, 2, 3, 6	2.60 (1.83) 1, 1, 2, 3, 6
	2	2.02 (0.87) 1, 1, 2, 2, 4	1.85 (0.83) 1, 1, 2, 2, 3	1.73 (0.82) 1, 1, 2, 2, 3	1.68 (0.84) 1, 1, 1, 2, 3	1.64 (0.90) 1, 1, 1, 2, 3
2.00	0	7.27 (4.29) 2, 4, 6, 9, 16	6.95 (4.57) 2, 4, 6, 9, 16	7.27 (5.61) 2, 4, 6, 9, 18	7.86 (6.57) 2, 3, 6, 10, 21	9.92 (9.54) 1, 4, 7, 13, 29
	0.25	7.02 (4.01) 2, 4, 6, 9, 15	6.64 (4.16) 2, 4, 6, 8, 15	6.88 (5.11) 2, 3, 6, 9, 17	7.53 (6.25) 2, 3, 6, 10, 20	9.50(8.94) 1, 3, 7, 12, 27
	0.5	6.24 (3.54) 2, 4, 5, 8, 13	5.91 (3.91) 2, 3, 5, 7, 13	6.07 (4.43) 1, 3, 5, 8, 15	6.35 (5.18) 1, 3, 5, 8, 16	7.75 (7.17) 1, 3, 6, 10, 22
	0.75	5.29 (2.92) 2, 3, 5, 7, 11	4.96 (2.98) 2, 3, 4, 6, 11	4.94 (3.46) 1, 3, 4, 6, 11	5.12 (3.83) 1, 2, 4, 7, 13	5.88 (5.09) 1, 2, 4, 8, 16
	1	4.44 (2.40) 2, 3, 4, 6, 9	4.08 (2.37) 1, 2, 4, 5, 9	3.99 (2.62) 1, 2, 3, 5, 9	4.11 (3.02) 1, 2, 3, 5, 10	4.43 (3.64) 1, 2, 3, 6, 12
	1.5	3.14 (1.55) 1, 2, 3, 4, 6	2.85 (1.49) 1, 2, 3, 4, 6	2.70 (1.59) 1, 2, 2, 3, 6	2.68 (1.70) 1, 1, 2, 3, 6	2.76 (1.96) 1, 1, 2, 4, 7
	2	2.34 (1.05) 1, 2, 2, 3, 4	2.12 (1.02) 1, 1, 2, 3, 4	1.98 (1.02) 1, 1, 2, 2, 4	1.91 (1.04) 1, 1, 2, 2, 4	1.92 (1.17) 1, 1, 2, 2, 4

,

Table A2. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Exp\left(\lambda \right)$ ($\lambda =1.5$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	149.36 (255.23) 10, 28, 63, 152, 581	174.33 (278.04) 9, 30, 77, 191, 666	210.38 (315.29) 8, 36, 95, 242, 796	225.28 (320.25) 8, 39, 109, 272, 855	238.07 (316.87) 9, 47, 125, 291, 870
	0.25	90.40 (166.60) 8, 21, 42, 90, 318	107.87 (182.69) 7, 22, 50, 117, 385	133.25 (207.74) 6, 25, 63, 151, 497	147.38 (224.70) 6, 28, 71, 171, 555	157.32 (220.45) 6, 32, 84, 194, 544
	0.5	27.18 (34.89) 5, 10, 18, 31, 76	31.18 (45.97) 4, 10, 18, 35, 98	42.15 (69.63) 3, 10, 21, 48, 147	47.85 (77.21) 3, 10, 25, 56, 166	57.12 (84.40) 3, 12, 30, 68, 202
	0.75	10.16 (7.89) 3, 5, 8, 13, 25	10.20 (9.40) 2, 5, 8, 12, 27	11.52 (12.57) 2, 4, 8, 14, 33	13.55 (17.35) 2, 4, 8, 16, 41	17.44 (22.82) 2, 5, 10, 21, 57
	1	5.27 (3.09) 2, 3, 5, 7, 11	5.01 (3.27) 1, 3, 4, 6, 11	5.06 (3.87) 1, 3, 4, 6, 12	5.38 (4.68) 1, 2, 4, 7, 14	6.51 (6.96) 1, 2, 4, 8, 19
	1.5	2.45 (1.06) 1, 2, 2, 3, 4	2.23 (1.04) 1, 2, 2, 3, 4	2.09 (1.08) 1, 1, 2, 3, 4	2.05 (1.12) 1, 1, 2, 3, 4	2.08 (1.31) 1, 1, 2, 3, 5
	2	1.60 (0.62) 1, 1, 2, 2, 3	1.46 (0.58) 1, 1, 1, 2, 2	1.35 (0.54) 1, 1, 1, 2, 2	1.32 (0.54) 1, 1, 1, 2, 2	1.27 (0.53) 1, 1, 1, 1, 2
1.25	0	45.30 (64.24) 7, 15, 28, 52, 136	53.26 (73.46) 5, 15, 31, 62, 175	65.08 (88.89) 5, 16, 36, 78, 215	73.94 (96.64) 4, 18, 42, 93, 253	89.34 (112.93) 4, 22, 53, 114, 287
	0.25	32.61 (40.04) 5, 12, 22, 38, 96	36.95 (48.50) 5, 12, 23, 44, 111	45.12 (59.63) 4, 12, 26, 54, 150	52.49 (68.70) 4, 13, 31, 65, 173	62.92 (79.60) 3, 15, 37, 80, 211
	0.5	16.37 (14.65) 3, 8, 12, 20, 42	17.44 (18.55) 3, 7, 12, 21, 49	20.21 (23.81) 2, 7, 13, 25, 61	22.85 (28.33) 2, 7, 14, 28, 73	28.44 (35.45) 2, 7, 17, 36, 93
	0.75	8.85 (6.25) 2, 5, 7, 11, 20	8.57 (6.79) 2, 4, 7, 11, 21	9.09 (8.58) 2, 4, 7, 11, 25	10.06 (10.60) 1, 4, 7, 13, 29	12.28 (13.50) 1, 4, 8, 16, 39
	1	5.34 (3.13) 2, 3, 5, 7, 11	5.05 (3.30) 1, 3, 4, 6, 11	5.02 (3.87) 1, 3, 4, 6, 12	5.21 (4.32) 1, 2, 4, 7, 13	6.09 (5.97) 1, 2, 4, 8, 17
	1.5	2.73 (1.28) 1, 2, 2, 3, 5	2.52 (1.26) 1, 2, 2, 3, 5	2.37 (1.30) 1, 1, 2, 3, 5	2.33 (1.36) 1, 1, 2, 3, 5	2.43 (1.71) 1, 1, 2, 3, 6
	2	1.83 (0.75) 1, 1, 2, 2, 3	1.68 (0.73) 1, 1, 2, 2, 3	1.55 (0.69) 1, 1, 1, 2, 3	1.50 (0.68) 1, 1, 1, 2, 3	1.48 (0.76) 1, 1, 1, 2, 3
1.5	0	17.51 (14.73) 4, 8, 14, 22, 44	18.53 (17.88) 3, 8, 13, 23, 51	22.05 (23.91) 3, 8, 15, 28, 66	25.44 (28.76) 3, 8, 17, 33, 78	33.97 (38.22) 3, 10, 22, 44, 107
	0.25	15.11 (11.85) 4, 7, 12, 19, 37	15.63 (14.41) 3, 7, 12, 20, 41	18.19 (19.00) 3, 7, 12, 23, 55	21.13 (23.89) 2, 7, 14, 27, 65	27.21 (31.25) 2, 8, 17, 35, 86
	0.5	10.85 (7.88) 3, 6, 9, 14, 25	10.76 (8.89) 2, 5, 8, 14, 27	11.79 (11.32) 2, 5, 8, 15, 33	13.06 (14.06) 2, 5, 9, 17, 37	16.17 (17.52) 2, 5, 11, 21, 49
	0.75	7.34 (4.57) 2, 4, 6, 9, 16	7.03 (4.97) 2, 4, 6, 9, 16	7.25 (6.12) 1, 3, 6, 9, 19	7.63 (6.96) 1, 3, 6, 10, 21	9.00 (8.76) 1, 3, 6, 12, 26
	1	5.14 (2.98) 2, 3, 4, 7, 11	4.79 (3.01) 1, 3, 4, 6, 10	4.75 (3.50) 1, 2, 4, 6, 11	4.86 (3.80) 1, 2, 4, 6, 12	5.45 (4.80) 1, 2, 4, 7, 15
	1.5	2.94 (1.44) 1, 2, 3, 4, 6	2.70 (1.40) 1, 2, 2, 3, 5	2.53 (1.45) 1, 2, 2, 3, 5	2.53 (1.63) 1, 1, 2, 3, 6	2.59 (1.82) 1, 1, 2, 3, 6
	2	2.02 (0.87) 1, 1, 2, 2, 4	1.85 (0.83) 1, 1, 2, 2, 3	1.72 (0.83) 1, 1, 2, 2, 3	1.66 (0.83) 1, 1, 1, 2, 3	1.65 (0.92) 1, 1, 1, 2, 3
2	0	7.24 (4.18) 2, 4, 6, 9, 15	6.85 (4.31) 2, 4, 6, 9, 15	7.18 (5.55) 2, 3, 6, 9, 18	7.74 (6.39) 2, 3, 6, 10, 20	9.74 (9.22) 1, 4, 7, 13, 28
	0.25	6.89 (3.88) 2, 4, 6, 9, 14	6.59 (4.23) 2, 4, 6, 8, 15	6.79 (5.07) 2, 3, 5, 9, 16	7.34 (6.17) 1, 3, 6, 9, 19	9.09 (8.41) 1, 3, 7, 12, 25
	0.5	6.14 (3.46) 2, 4, 5, 8, 13	5.89 (3.69) 2, 3, 5, 8, 13	5.86 (4.27) 1, 3, 5, 8, 14	6.29 (5.08) 1, 3, 5, 8, 16	7.42 (6.84) 1, 3, 5, 10, 21
	0.75	5.27 (2.90) 2, 3, 5, 7, 11	4.93 (2.90) 2, 3, 4, 6, 10	4.93 (3.37) 1, 3, 4, 6, 12	5.11 (3.90) 1, 2, 4, 7, 13	5.81 (5.02) 1, 2, 4, 8, 16
	1	4.39 (2.35) 2, 3, 4, 6, 9	4.12 (2.40) 1, 2, 4, 5, 9	3.94 (2.57) 1, 2, 3, 5, 9	4.03 (2.93) 1, 2, 3, 5, 10	4.43 (3.57) 1, 2, 3, 6, 11
	1.5	3.11 (1.51) 1, 2, 3, 4, 6	2.87 (1.49) 1, 2, 3, 4, 6	2.71 (1.56) 1, 2, 2, 3, 6	2.68 (1.69) 1, 1, 2, 3, 6	2.75 (1.97) 1, 1, 2, 4, 7
	2	2.33 (1.04) 1, 2, 2, 3, 4	2.14 (1.02) 1, 1, 2, 3, 4	1.96 (1.02) 1, 1, 2, 2, 4	1.93 (1.07) 1, 1, 2, 2, 4	1.91 (1.14) 1, 1, 2, 2, 4

and

Table A3. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Exp\left(\lambda \right)$ ($\lambda =2$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	29.89 (49.23) 5, 11, 18, 32, 87	38.99 (70.81) 4, 10, 19, 40, 134	56.32 (100.75) 4, 11, 25, 60, 209	73.25 (133.33) 4, 13, 31, 78, 274	91.77 (150.44) 4, 16, 43, 105, 337
	0.25	25.22 (31.68) 5, 10, 17, 29, 72	30.99 (44.89) 4, 10, 18, 34, 102	45.54 (76.94) 4, 10, 23, 50, 159	56.51 (91.59) 3, 11, 27, 64, 203	71.71 (111.56) 3, 14, 36, 84, 258
	0.5	15.99 (13.94) 4, 8, 12, 20, 39	17.29 (18.41) 3, 7, 12, 21, 48	23.15 (32.88) 3, 7, 14, 27, 74	27.24 (37.24) 2, 7, 15, 33, 88	35.79 (48.60) 2, 9, 20, 44, 123
	0.75	8.82 (5.83) 3, 5, 7, 11, 20	8.72 (7.06) 2, 4, 7, 11, 21	9.62 (9.50) 2, 4, 7, 12, 26	11.22 (13.25) 2, 4, 7, 14, 33	14.53 (18.58) 2, 4, 9, 18, 46
	1	5.14 (2.87) 2, 3, 5, 6, 11	4.81 (3.04) 1, 3, 4, 6, 10	4.78 (3.54) 1, 2, 4, 6, 11	5.15 (4.34) 1, 2, 4, 6, 13	6.15 (6.37) 1, 2, 4, 8, 18
	1.5	2.47 (1.07) 1, 2, 2, 3, 4	2.23 (1.04) 1, 2, 2, 3, 4	2.08 (1.07) 1, 1, 2, 3, 4	2.04 (1.13) 1, 1, 2, 2, 4	2.07 (1.30) 1, 1, 2, 3, 4
	2	1.59 (0.62) 1, 1, 2, 2, 3	1.45 (0.58) 1, 1, 1, 2, 2	1.35 (0.53) 1, 1, 1, 2, 2	1.32 (0.53) 1, 1, 1, 2, 2	1.28 (0.54) 1, 1, 1, 1, 2
1.25	0	20.46 (21.13) 4, 9, 15, 25, 53	22.89 (27.26) 4, 9, 15, 27, 66	31.08 (39.50) 3, 9, 18, 38, 101	39.85 (55.53) 3, 10, 22, 48, 133	49.75 (65.55) 3, 11, 27, 62, 174
	0.25	17.68 (15.85) 4, 9, 14, 22, 44	19.97 (21.94) 3, 8, 14, 24, 56	24.70 (29.76) 3, 8, 15, 30, 77	30.40 (38.97) 3, 8, 18, 38, 99	39.90 (52.06) 3, 10, 23, 50, 131
	0.5	12.42 (9.11) 3, 6, 10, 16, 29	12.74 (11.35) 3, 6, 10, 16, 33	14.82 (16.36) 2, 6, 10, 18, 43	17.55 (20.88) 2, 6, 11, 22, 53	21.84 (26.51) 2, 6, 13, 27, 71
	0.75	7.87 (5.00) 2, 4, 7, 10, 17	7.67 (5.72) 2, 4, 6, 10, 18	7.97 (6.88) 2, 4, 6, 10, 21	8.85 (8.59) 1, 4, 6, 11, 24	10.81 (11.33) 1, 3, 7, 14, 33
	1	5.19 (2.91) 2, 3, 5, 7, 11	4.83 (3.03) 1, 3, 4, 6, 10	4.82 (3.53) 1, 2, 4, 6, 11	4.95 (4.02) 1, 2, 4, 6, 13	5.59 (5.25) 1, 2, 4, 7, 15
	1.5	2.74 (1.29) 1, 2, 3, 3, 5	2.52 (1.25) 1, 2, 2, 3, 5	2.34 (1.29) 1, 1, 2, 3, 5	2.31 (1.36) 1, 1, 2, 3, 5	2.34 (1.57) 1, 1, 2, 3, 5
	2	1.82 (0.75) 1, 1, 2, 2, 3	1.67 (0.72) 1, 1, 2, 2, 3	1.54 (0.69) 1, 1, 1, 2, 3	1.50 (0.68) 1, 1, 1, 2, 3	1.46 (0.72) 1, 1, 1, 2, 3
1.5	0	12.96 (9.29) 3, 7, 11, 16, 30	13.51 (11.34) 3, 6, 10, 17, 34	15.62 (15.93) 2, 6, 11, 19, 44	19.01 (20.68) 2, 6, 12, 24, 57	25.14 (29.48) 2, 7, 16, 32, 79
	0.25	11.77 (8.04) 3, 6, 10, 15, 27	12.04 (9.56) 3, 6, 10, 15, 30	13.87 (13.28) 2, 5, 10, 18, 38	16.37 (17.34) 2, 6, 11, 21, 49	21.09 (23.76) 2, 6, 13, 27, 66
	0.5	9.28 (5.89) 3, 5, 8, 12, 20	9.25 (6.84) 2, 5, 8, 12, 22	10.01 (8.87) 2, 4, 8, 13, 27	11.16 (11.08) 2, 4, 8, 14, 32	14.11 (15.18) 1, 5, 9, 18, 43
	0.75	6.76 (4.08) 2, 4, 6, 9, 14	6.44 (4.31) 2, 3, 5, 8, 15	6.56 (5.25) 1, 3, 5, 8, 16	7.10 (6.16) 1, 3, 5, 9, 19	8.43 (8.30) 1, 3, 6, 11, 25
	1	4.94 (2.73) 2, 3, 4, 6, 10	4.61 (2.82) 1, 3, 4, 6, 10	4.56 (3.19) 1, 2, 4, 6, 11	4.62 (3.58) 1, 2, 4, 6, 12	5.08 (4.34) 1, 2, 4, 7, 14
	1.5	2.92 (1.41) 1, 2, 3, 4, 6	2.66 (1.36) 1, 2, 2, 3, 5	2.54 (1.45) 1, 2, 2, 3, 5	2.49 (1.53) 1, 1, 2, 3, 5	2.54 (1.82) 1, 1, 2, 3, 6
	2	2.04 (0.88) 1, 1, 2, 2, 4	1.88 (0.83) 1, 1, 2, 2, 3	1.71 (0.81) 1, 1, 2, 2, 3	1.67 (0.84) 1, 1, 1, 2, 3	1.64 (0.89) 1, 1, 1, 2, 3
2	0	6.74 (3.64) 2, 4, 6, 9, 14	6.41 (3.97) 2, 4, 6, 8, 14	6.55 (4.82) 2, 3, 5, 8, 16	7.17 (5.80) 1, 3, 6, 9, 18	8.88 (8.23) 1, 3, 6, 12, 25
	0.25	6.54 (3.54) 2, 4, 6, 8, 13	6.22 (3.86) 2, 4, 5, 8, 13	6.30 (4.54) 2, 3, 5, 8, 15	6.82 (5.56) 1, 3, 5, 9, 18	8.29 (7.65) 1, 3, 6, 11, 23
	0.5	5.91 (3.19) 2, 4, 5, 7, 12	5.54 (3.34) 2, 3, 5, 7, 12	5.63 (3.98) 1, 3, 5, 7, 13	5.89 (4.52) 1, 3, 5, 8, 15	6.98 (6.30) 1, 3, 5, 9, 19
	0.75	5.08 (2.70) 2, 3, 5, 6, 10	4.79 (2.80) 1, 3, 4, 6, 10	4.74 (3.23) 1, 2, 4, 6, 11	4.89 (3.65) 1, 2, 4, 6, 12	5.63 (4.86) 1, 2, 4, 7, 15
	1	4.34 (2.25) 2, 3, 4, 5, 9	4.01 (2.27) 1, 2, 4, 5, 8	3.89 (2.47) 1, 2, 3, 5, 9	3.93 (2.85) 1, 2, 3, 5, 9	4.34 (3.56) 1, 2, 3, 6, 11
	1.5	3.12 (1.52) 1, 2, 3, 4, 6	2.84 (1.48) 1, 2, 3, 4, 6	2.64 (1.51) 1, 2, 2, 3, 6	2.64 (1.65) 1, 1, 2, 3, 6	2.69 (1.87) 1, 1, 2, 3, 6
	2	2.33 (1.05) 1, 2, 2, 3, 4	2.11 (1.02) 1, 1, 2, 3, 4	1.97 (1.02) 1, 1, 2, 2, 4	1.89 (1.05) 1, 1, 2, 2, 4	1.90 (1.16) 1, 1, 2, 2, 4

. In these tables, the first row of each cell shows the $ARL$ value, with the SDRL given in parentheses. The second row presents the 5th, 25th, 50th, 75th, and 95th percentiles.

Based on Table A1, Table A2, Table A3,

Table A4. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Weibull({\alpha }_1,{\beta }_1)$ (${\alpha }_1=1$, ${\beta }_1=1$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	376.86 (480.80) 17, 71, 187, 468, 1627	378.51 (462.28) 17, 75, 199, 485, 1492	373.73 (448.92) 17, 82, 210, 492, 1448	371.96 (433.03) 16, 79, 210, 491, 1350	366.58 (419.88) 15, 83, 216, 481, 1294
	0.25	180.58 (291.41) 10, 33, 79, 193, 687	197.65 (298.47) 9, 35, 91, 222, 780	216.42 (304.94) 8, 39, 108, 257, 801	219.02 (299.70) 8, 42, 113, 265, 825	228.03 (300.01) 8, 46, 120, 289, 815
	0.5	33.27 (52.15) 5, 11, 20, 37, 103	40.04 (64.79) 4, 11, 21, 44, 132	50.54 (86.81) 4, 11, 25, 56, 176	59.55 (103.85) 3, 12, 29, 66, 208	71.05 (108.87) 3, 14, 36, 83, 252
	0.75	10.49 (8.43) 3, 5, 8, 13, 25	10.79 (10.97) 2, 5, 8, 13, 28	12.30 (14.79) 2, 4, 8, 15, 36	14.43 (20.47) 2, 5, 8, 17, 46	18.76 (26.06) 2, 5, 11, 22, 62
	1	5.33 (3.15) 2, 3, 5, 7, 11	5.08 (3.41) 1, 3, 4, 6, 11	5.10 (3.99) 1, 3, 4, 6, 12	5.48 (4.86) 1, 2, 4, 7, 14	6.60 (7.26) 1, 2, 4, 8, 19
	1.5	2.45 (1.06) 1, 2, 2, 3, 4	2.26 (1.06) 1, 2, 2, 3, 4	2.09 (1.07) 1, 1, 2, 3, 4	2.05 (1.13) 1, 1, 2, 3, 4	2.10 (1.36) 1, 1, 2, 3, 5
	2	1.59 (0.61) 1, 1, 2, 2, 3	1.45 (0.58) 1, 1, 1, 2, 2	1.35 (0.54) 1, 1, 1, 2, 2	1.31 (0.52) 1, 1, 1, 2, 2	1.28 (0.54) 1, 1, 1, 1, 2
1.25	0	64.99 (97.92) 7, 19, 36, 73, 209	71.04 (99.27) 6, 19, 40, 84, 242	85.34 (114.89) 5, 20, 48, 106, 289	94.67 (120.96) 5, 22, 54, 119, 325	110.65 (138.05) 5, 26, 65, 140, 371
	0.25	42.71 (58.64) 6, 14, 26, 49, 133	46.72 (64.13) 5, 13, 27, 56, 152	55.47 (76.99) 4, 14, 31, 66, 185	62.98 (83.63) 4, 15, 36, 78, 215	75.75 (97.99) 4, 18, 43, 97, 254
	0.5	17.91 (17.62) 4, 8, 13, 22, 47	19.09 (21.85) 3, 7, 13, 23, 56	21.76 (26.50) 3, 7, 14, 26, 67	25.61 (33.42) 2, 7, 15, 31, 84	31.88 (41.32) 2, 8, 18, 40, 103
	0.75	9.06 (6.53) 2, 5, 7, 12, 21	8.70 (7.04) 2, 4, 7, 11, 21	9.51 (9.14) 2, 4, 7, 12, 26	10.72 (11.57) 2, 4, 7, 13, 31	13.03 (15.92) 1, 4, 8, 16, 41
	1	5.35 (3.20) 2, 3, 5, 7, 11	5.09 (3.30) 1, 3, 4, 6, 11	5.09 (3.88) 1, 3, 4, 6, 12	5.31 (4.50) 1, 2, 4, 7, 14	6.07 (5.72) 1, 2, 4, 8, 17
	1.5	2.74 (1.29) 1, 2, 3, 3, 5	2.51 (1.26) 1, 2, 2, 3, 5	2.37 (1.31) 1, 1, 2, 3, 5	2.34 (1.38) 1, 1, 2, 3, 5	2.38 (1.65) 1, 1, 2, 3, 6
	2	1.83 (0.75) 1, 1, 2, 2, 3	1.66 (0.71) 1, 1, 2, 2, 3	1.54 (0.68) 1, 1, 1, 2, 3	1.50 (0.70) 1, 1, 1, 2, 3	1.46 (0.71) 1, 1, 1, 2, 3
1.50	0	18.73 (15.69) 4, 9, 15, 24, 47	20.10 (19.51) 3, 8, 14, 25, 56	24.51 (27.79) 3, 8, 16, 30, 75	28.41 (31.97) 3, 9, 18, 36, 89	37.17 (43.17) 3, 10, 23, 48, 117
	0.25	16.10 (13.03) 4, 8, 13, 20, 40	16.81 (15.59) 3, 7, 12, 21, 46	19.82 (21.26) 3, 7, 13, 25, 58	23.03 (26.76) 2, 7, 15, 29, 72	29.15 (32.75) 2, 8, 19, 38, 92
	0.5	11.20 (8.10) 3, 6, 9, 14, 26	11.03 (9.23) 2, 5, 9, 14, 28	12.25 (12.00) 2, 5, 9, 15, 34	13.98 (14.92) 2, 5, 9, 18, 41	17.21 (19.41) 2, 5, 11, 22, 53
	0.75	7.45 (4.74) 2, 4, 6, 10, 16	7.11 (5.22) 2, 4, 6, 9, 17	7.27 (5.99) 1, 3, 6, 9, 19	7.98 (7.27) 1, 3, 6, 10, 22	9.48 (10.11) 1, 3, 6, 12, 28
	1	5.13 (3.02) 2, 3, 4, 6, 11	4.78 (3.05) 1, 3, 4, 6, 10	4.79 (3.50) 1, 2, 4, 6, 12	4.98 (3.87) 1, 2, 4, 6, 12	5.56 (5.03) 1, 2, 4, 7, 15
	1.5	2.94 (1.45) 1, 2, 3, 4, 6	2.69 (1.41) 1, 2, 2, 3, 5	2.55 (1.46) 1, 2, 2, 3, 5	2.52 (1.58) 1, 1, 2, 3, 6	2.58 (1.79) 1, 1, 2, 3, 6
	2	2.02 (0.88) 1, 1, 2, 2, 4	1.85 (0.84) 1, 1, 2, 2, 3	1.73 (0.83) 1, 1, 2, 2, 3	1.67 (0.82) 1, 1, 1, 2, 3	1.64 (0.89) 1, 1, 1, 2, 3
2.00	0	7.25 (4.21) 2, 4, 6, 9, 15	6.98 (4.55) 2, 4, 6, 9, 16	7.29 (5.56) 2, 4, 6, 9, 18	7.95 (6.69) 2, 3, 6, 10, 21	10.19 (10.20) 1, 4, 7, 13, 29
	0.25	7.00 (4.09) 2, 4, 6, 9, 15	6.72 (4.40) 2, 4, 6, 9, 15	6.94 (5.28) 2, 3, 6, 9, 17	7.64 (6.38) 1, 3, 6, 10, 20	9.31 (8.63) 1, 3, 7, 12, 26
	0.5	6.28 (3.61) 2, 4, 6, 8, 13	5.88 (3.74) 2, 3, 5, 8, 13	5.97 (4.38) 1, 3, 5, 8, 14	6.32 (5.05) 1, 3, 5, 8, 16	7.61 (6.96) 1, 3, 6, 10, 21
	0.75	5.29 (2.94) 2, 3, 5, 7, 11	4.95 (2.99) 2, 3, 4, 6, 11	4.94 (3.42) 1, 3, 4, 6, 12	5.08 (3.89) 1, 2, 4, 7, 13	5.96 (5.30) 1, 2, 4, 8, 16
	1	4.38 (2.26) 2, 3, 4, 6, 9	4.06 (2.36) 1, 2, 4, 5, 8	3.99 (2.63) 1, 2, 3, 5, 9	4.12 (2.98) 1, 2, 3, 5, 10	4.48 (3.62) 1, 2, 3, 6, 12
	1.5	3.11 (1.54) 1, 2, 3, 4, 6	2.85 (1.48) 1, 2, 3, 4, 6	2.69 (1.58) 1, 2, 2, 3, 6	2.69 (1.73) 1, 1, 2, 3, 6	2.78 (2.00) 1, 1, 2, 4, 7
	2	2.33 (1.05) 1, 2, 2, 3, 4	2.13 (1.02) 1, 1, 2, 3, 4	1.99 (1.03) 1, 1, 2, 2, 4	1.92 (1.06) 1, 1, 2, 2, 4	1.93 (1.18) 1, 1, 2, 2, 4

,

Table A5. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Weibull({\alpha }_1,{\beta }_1)$ (${\alpha }_1=1.5$, ${\beta }_1=1$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	42.34 (65.29) 6, 13, 24, 47, 132	47.17 (70.52) 5, 12, 25, 53, 159	57.32 (82.60) 4, 13, 30, 67, 200	65.19 (95.03) 4, 14, 34, 78, 233	73.24 (104.99) 4, 16, 40, 89, 251
	0.25	34.69 (44.13) 5, 12, 22, 40, 104	38.23 (50.81) 4, 11, 23, 45, 126	47.16 (68.06) 4, 12, 27, 56, 157	52.44 (75.54) 3, 12, 29, 63, 179	59.43 (76.97) 3, 14, 35, 75, 198
	0.5	19.21 (19.96) 4, 9, 14, 23, 51	20.35 (23.12) 3, 8, 14, 25, 58	24.20 (29.50) 3, 7, 15, 30, 77	27.26 (35.06) 2, 7, 16, 34, 91	33.25 (42.02) 2, 8, 19, 42, 111
	0.75	9.18 (7.33) 3, 5, 8, 12, 21	9.12 (8.00) 2, 4, 7, 11, 23	9.93 (9.64) 2, 4, 7, 12, 27	11.22 (12.26) 2, 4, 8, 14, 33	14.11 (16.55) 1, 4, 9, 18, 44
	1	5.16 (2.91) 2, 3, 5, 6, 11	4.78 (2.97) 1, 3, 4, 6, 10	4.84 (3.79) 1, 2, 4, 6, 11	5.02 (4.18) 1, 2, 4, 6, 13	6.05 (6.34) 1, 2, 4, 8, 18
	1.5	2.44 (1.06) 1, 2, 2, 3, 4	2.24 (1.04) 1, 2, 2, 3, 4	2.08 (1.07) 1, 1, 2, 3, 4	2.05 (1.15) 1, 1, 2, 3, 4	2.08 (1.31) 1, 1, 2, 3, 5
	2	1.61 (0.62) 1, 1, 2, 2, 3	1.45 (0.58) 1, 1, 1, 2, 2	1.35 (0.55) 1, 1, 1, 2, 2	1.32 (0.53) 1, 1, 1, 2, 2	1.27 (0.52) 1, 1, 1, 1, 2
1.25	0	25.77 (28.11) 5, 11, 18, 31, 70	28.56 (33.54) 4, 10, 18, 35, 89	33.66 (39.95) 3, 10, 21, 43, 106	37.12 (44.25) 3, 10, 23, 47, 120	45.92 (56.47) 3, 11, 28, 59, 149
	0.25	21.52 (20.11) 4, 9, 16, 26, 57	22.53 (23.12) 3, 9, 16, 28, 64	27.03 (33.07) 3, 8, 17, 33, 83	30.54 (36.80) 3, 9, 19, 38, 99	36.88 (45.74) 3, 10, 23, 47, 116
	0.5	13.74 (11.07) 3, 7, 11, 17, 33	13.98 (13.11) 3, 6, 10, 17, 36	15.58 (17.26) 2, 6, 11, 20, 45	17.51 (20.14) 2, 6, 11, 22, 53	21.04 (24.04) 2, 6, 13, 27, 66
	0.75	8.13 (5.44) 2, 4, 7, 10, 18	7.92 (6.09) 2, 4, 6, 10, 19	8.19 (7.17) 2, 4, 6, 10, 21	8.90 (8.65) 1, 3, 6, 11, 25	10.91 (11.38) 1, 3, 7, 14, 33
	1	5.16 (2.94) 2, 3, 5, 7, 11	4.84 (3.07) 1, 3, 4, 6, 11	4.73 (3.42) 1, 2, 4, 6, 11	5.04 (4.18) 1, 2, 4, 6, 13	5.65 (5.29) 1, 2, 4, 7, 16
	1.5	2.74 (1.28) 1, 2, 3, 3, 5	2.49 (1.25) 1, 2, 2, 3, 5	2.35 (1.29) 1, 1, 2, 3, 5	2.33 (1.37) 1, 1, 2, 3, 5	2.33 (1.56) 1, 1, 2, 3, 5
	2	1.83 (0.75) 1, 1, 2, 2, 3	1.67 (0.70) 1, 1, 2, 2, 3	1.55 (0.69) 1, 1, 1, 2, 3	1.49 (0.68) 1, 1, 1, 2, 3	1.46 (0.71) 1, 1, 1, 2, 3
1.5	0	14.24 (10.27) 4, 7, 12, 18, 33	14.92 (13.17) 3, 7, 11, 19, 39	17.01 (16.75) 3, 6, 12, 22, 48	19.40 (20.32) 2, 6, 13, 25, 58	24.38 (27.03) 2, 7, 16, 32, 75
	0.25	12.92 (9.60) 3, 7, 10, 16, 30	12.73 (10.40) 3, 6, 10, 16, 32	14.44 (13.81) 2, 6, 10, 19, 41	16.35 (16.55) 2, 6, 11, 21, 48	20.50 (22.33) 2, 6, 13, 27, 64
	0.5	9.71 (6.53) 3, 5, 8, 12, 22	9.61 (7.22) 2, 5, 8, 12, 23	10.29 (9.34) 2, 4, 8, 13, 27	11.22 (10.98) 2, 4, 8, 14, 32	13.60 (14.11) 1, 4, 9, 18, 41
	0.75	6.95 (4.21) 2, 4, 6, 9, 15	6.63 (4.60) 2, 4, 6, 9, 15	6.77 (5.50) 1, 3, 5, 9, 17	7.17 (6.38) 1, 3, 5, 9, 19	8.42 (8.34) 1, 3, 6, 11, 25
	1	4.96 (2.83) 2, 3, 4, 6, 10	4.69 (2.93) 1, 3, 4, 6, 10	4.59 (3.22) 1, 2, 4, 6, 11	4.73 (3.66) 1, 2, 4, 6, 12	5.25 (4.67) 1, 2, 4, 7, 15
	1.5	2.92 (1.42) 1, 2, 3, 4, 6	2.71 (1.41) 1, 2, 2, 3, 5	2.52 (1.43) 1, 2, 2, 3, 5	2.48 (1.53) 1, 1, 2, 3, 5	2.57 (1.81) 1, 1, 2, 3, 6
	2	2.03 (0.87) 1, 1, 2, 2, 4	1.86 (0.83) 1, 1, 2, 2, 3	1.70 (0.80) 1, 1, 2, 2, 3	1.66 (0.83) 1, 1, 1, 2, 3	1.64 (0.87) 1, 1, 1, 2, 3
2	0	6.86 (3.81) 2, 4, 6, 9, 14	6.59 (4.23) 2, 4, 6, 8, 15	6.81 (4.99) 2, 3, 6, 9, 16	7.26 (5.99) 1, 3, 6, 10, 19	8.93 (8.24) 1, 3, 6, 12, 25
	0.25	6.61 (3.62) 2, 4, 6, 8, 13	6.22 (3.93) 2, 3, 5, 8, 14	6.39 (4.74) 2, 3, 5, 8, 16	6.90 (5.59) 1, 3, 5, 9, 18	8.28 (7.57) 1, 3, 6, 11, 23
	0.5	5.96 (3.26) 2, 4, 5, 8, 12	5.58 (3.41) 2, 3, 5, 7, 12	5.69 (4.06) 1, 3, 5, 7, 13	5.91 (4.60) 1, 3, 5, 8, 15	7.09 (6.39) 1, 3, 5, 9, 20
	0.75	5.12 (2.79) 2, 3, 5, 6, 10	4.77 (2.82) 1, 3, 4, 6, 10	4.68 (3.17) 1, 2, 4, 6, 11	4.86 (3.58) 1, 2, 4, 6, 12	5.56 (4.73) 1, 2, 4, 7, 15
	1	4.32 (2.24) 2, 3, 4, 5, 8	3.96 (2.26) 1, 2, 3, 5, 8	3.90 (2.56) 1, 2, 3, 5, 9	3.93 (2.81) 1, 2, 3, 5, 9	4.35 (3.54) 1, 2, 3, 6, 11
	1.5	3.07 (1.49) 1, 2, 3, 4, 6	2.84 (1.50) 1, 2, 3, 4, 6	2.67 (1.55) 1, 2, 2, 3, 6	2.65 (1.67) 1, 1, 2, 3, 6	2.73 (1.93) 1, 1, 2, 4, 6
	2	2.33 (1.03) 1, 2, 2, 3, 4	2.12 (1.01) 1, 1, 2, 3, 4	1.96 (0.99) 1, 1, 2, 2, 4	1.92 (1.05) 1, 1, 2, 2, 4	1.90 (1.13) 1, 1, 2, 2, 4

and

Table A6. The OOC performance of the EWMA-Lepage scheme for $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Weibull({\alpha }_1,{\beta }_1)$ (${\alpha }_1=2$, ${\beta }_1=1$, $m=100$, $n=5$, $AR{L}_0=370$).

a	b	$\mathit{r}=0.05$	$\mathit{r}=0.1$	$\mathit{r}=0.2$	$\mathit{r}=0.3$	$\mathit{r}=0.5$
1	0	10.73 (8.16) 3, 5, 9, 13, 26	10.77 (9.51) 2, 5, 8, 14, 28	12.09 (12.18) 2, 5, 8, 15, 35	13.22 (15.54) 2, 4, 9, 16, 39	16.52 (19.10) 2, 5, 10, 21, 51
	0.25	10.55 (7.99) 3, 5, 9, 13, 25	10.41 (8.81) 2, 5, 8, 13, 27	11.24 (11.09) 2, 4, 8, 14, 32	12.82 (13.91) 2, 4, 8, 16, 39	15.32 (17.09) 1, 5, 10, 19, 48
	0.5	9.11 (6.24) 2, 5, 8, 12, 21	8.79 (6.89) 2, 4, 7, 11, 21	9.47 (8.68) 2, 4, 7, 12, 25	10.47 (10.85) 2, 4, 7, 13, 30	12.58 (13.56) 1, 4, 8, 16, 38
	0.75	6.68 (3.98) 2, 4, 6, 8, 14	6.39 (4.25) 2, 3, 5, 8, 14	6.39 (5.06) 1, 3, 5, 8, 16	6.89 (6.40) 1, 3, 5, 9, 18	8.27 (8.47) 1, 3, 6, 11, 24
	1	4.59 (2.44) 2, 3, 4, 6, 9	4.25 (2.48) 1, 3, 4, 5, 9	4.07 (2.79) 1, 2, 3, 5, 9	4.27 (3.35) 1, 2, 3, 5, 10	4.86 (4.31) 1, 2, 4, 6, 13
	1.5	2.42 (1.03) 1, 2, 2, 3, 4	2.20 (1.01) 1, 2, 2, 3, 4	2.01 (0.99) 1, 1, 2, 2, 4	1.97 (1.06) 1, 1, 2, 2, 4	2.00 (1.23) 1, 1, 2, 2, 4
	2	1.60 (0.62) 1, 1, 2, 2, 3	1.45 (0.57) 1, 1, 1, 2, 2	1.34 (0.54) 1, 1, 1, 2, 2	1.29 (0.51) 1, 1, 1, 2, 2	1.26 (0.51) 1, 1, 1, 1, 2
1.25	0	10.05 (7.18) 3, 5, 8, 13, 24	9.70 (7.91) 2, 5, 8, 12, 25	10.62 (10.05) 2, 4, 8, 13, 29	11.81 (12.88) 2, 4, 8, 15, 35	14.59 (16.89) 1, 4, 9, 19, 45
	0.25	9.65 (6.66) 2, 5, 8, 12, 22	9.42 (7.51) 2, 5, 7, 12, 23	10.00 (9.71) 2, 4, 7, 13, 27	11.14 (11.67) 2, 4, 8, 14, 33	13.02 (14.42) 1, 4, 8, 17, 41
	0.5	8.13 (5.19) 2, 5, 7, 10, 18	7.79 (5.55) 2, 4, 6, 10, 18	8.03 (6.97) 2, 4, 6, 10, 21	8.73 (8.09) 1, 4, 6, 11, 24	10.45 (10.90) 1, 3, 7, 13, 31
	0.75	6.25 (3.61) 2, 4, 6, 8, 13	5.85 (3.80) 2, 3, 5, 8, 13	5.88 (4.47) 1, 3, 5, 8, 14	6.25 (5.42) 1, 3, 5, 8, 17	7.11 (6.99) 1, 3, 5, 9, 20
	1	4.59 (2.47) 2, 3, 4, 6, 9	4.24 (2.51) 1, 2, 4, 5, 9	4.15 (2.88) 1, 2, 3, 5, 9	4.19 (3.11) 1, 2, 3, 5, 10	4.58 (4.10) 1, 2, 3, 6, 12
	1.5	2.65 (1.22) 1, 2, 2, 3, 5	2.41 (1.18) 1, 2, 2, 3, 5	2.26 (1.19) 1, 1, 2, 3, 4	2.23 (1.29) 1, 1, 2, 3, 5	2.21 (1.43) 1, 1, 2, 3, 5
	2	1.81 (0.75) 1, 1, 2, 2, 3	1.65 (0.70) 1, 1, 2, 2, 3	1.52 (0.67) 1, 1, 1, 2, 3	1.47 (0.67) 1, 1, 1, 2, 3	1.43 (0.69) 1, 1, 1, 2, 3
1.5	0	8.43 (5.24) 2, 5, 7, 11, 18	8.12 (5.88) 2, 4, 7, 10, 19	8.46 (7.24) 2, 4, 7, 11, 22	9.32 (9.16) 1, 4, 7, 12, 26	11.05 (11.36) 1, 4, 7, 14, 32
	0.25	8.01 (4.90) 2, 5, 7, 10, 17	7.65 (5.39) 2, 4, 6, 10, 18	8.08 (6.64) 2, 4, 6, 10, 21	8.67 (7.94) 1, 4, 6, 11, 23	10.59 (10.83) 1, 4, 7, 14, 31
	0.5	7.04 (4.18) 2, 4, 6, 9, 15	6.58 (4.28) 2, 4, 6, 8, 15	6.79 (5.43) 1, 3, 5, 9, 17	7.05 (6.06) 1, 3, 5, 9, 19	8.42 (8.23) 1, 3, 6, 11, 25
	0.75	5.61 (3.13) 2, 3, 5, 7, 11	5.20 (3.26) 2, 3, 4, 7, 11	5.15 (3.75) 1, 3, 4, 7, 12	5.38 (4.33) 1, 2, 4, 7, 14	6.18 (5.57) 1, 2, 4, 8, 17
	1	4.45 (2.34) 2, 3, 4, 6, 9	4.11 (2.39) 1, 2, 4, 5, 9	3.96 (2.64) 1, 2, 3, 5, 9	3.98 (2.88) 1, 2, 3, 5, 9	4.35 (3.64) 1, 2, 3, 6, 11
	1.5	2.82 (1.34) 1, 2, 3, 4, 5	2.61 (1.31) 1, 2, 2, 3, 5	2.39 (1.32) 1, 1, 2, 3, 5	2.35 (1.40) 1, 1, 2, 3, 5	2.38 (1.59) 1, 1, 2, 3, 5
	2	2.00 (0.86) 1, 1, 2, 2, 4	1.83 (0.81) 1, 1, 2, 2, 3	1.68 (0.79) 1, 1, 2, 2, 3	1.62 (0.81) 1, 1, 1, 2, 3	1.59 (0.84) 1, 1, 1, 2, 3
2	0	5.76 (3.06) 2, 4, 5, 7, 11	5.32 (3.12) 2, 3, 5, 7, 11	5.24 (3.59) 1, 3, 4, 7, 12	5.49 (4.21) 1, 3, 4, 7, 14	6.43 (5.59) 1, 3, 5, 8, 17
	0.25	5.51 (2.86) 2, 3, 5, 7, 11	5.13 (3.02) 2, 3, 5, 7, 11	5.04 (3.37) 1, 3, 4, 7, 12	5.30 (3.98) 1, 3, 4, 7, 13	6.19 (5.41) 1, 3, 5, 8, 17
	0.5	5.12 (2.63) 2, 3, 5, 6, 10	4.77 (2.77) 2, 3, 4, 6, 10	4.65 (3.05) 1, 3, 4, 6, 11	4.75 (3.54) 1, 2, 4, 6, 12	5.48 (4.75) 1, 2, 4, 7, 15
	0.75	4.60 (2.40) 2, 3, 4, 6, 9	4.22 (2.38) 1, 3, 4, 5, 9	4.08 (2.62) 1, 2, 3, 5, 9	4.11 (2.97) 1, 2, 3, 5, 10	4.51 (3.62) 1, 2, 3, 6, 12
	1	4.01 (2.03) 1, 3, 4, 5, 8	3.64 (2.01) 1, 2, 3, 5, 8	3.50 (2.17) 1, 2, 3, 4, 8	3.48 (2.37) 1, 2, 3, 4, 8	3.77 (2.91) 1, 2, 3, 5, 9
	1.5	2.98 (1.41) 1, 2, 3, 4, 6	2.72 (1.40) 1, 2, 2, 3, 5	2.52 (1.44) 1, 1, 2, 3, 5	2.46 (1.49) 1, 1, 2, 3, 5	2.54 (1.73) 1, 1, 2, 3, 6
	2	2.27 (1.00) 1, 2, 2, 3, 4	2.07 (0.97) 1, 1, 2, 3, 4	1.90 (0.97) 1, 1, 2, 2, 4	1.86 (0.99) 1, 1, 2, 2, 4	1.84 (1.10) 1, 1, 2, 2, 4

, we compare the run length attributes of different distributions under different parameter settings. After comparing these distribution modes, we can see that the run length distributions of the proposed EWMA-Lepage monitoring scheme is still skewed to the right. The data in Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6 show that the $ARL$, $SDRL$, and percentiles of the distribution-free schemes decrease rapidly as the shift size in S or T increases. In the following analysis, we focus on the $ARL$ as a key performance metric to assess the detection capability of S or T under various shift magnitudes.

As shown in Table A1, Table A2 and Table A3, when S follows a normal distribution and T follows an exponential distribution, we consider the scenario where both the parameters of the exponential distribution and the scale parameter of the normal distribution remain unchanged. In this case, the monitoring scheme is more effective in detecting shifts in the location parameter. For example, as can be seen from Table A1, when $\lambda =1$, $r=0.05$, and $a=1$, the values of $AR{L}_1$ corresponding to $b=0.25$ and $b=0.5$ are 177.91 and 34.37, respectively. Similarly, when the location parameter of the normal distribution and the parameter of the exponential distribution remain unchanged, the scheme can also effectively detect shifts in the scale parameter of the normal distribution. As shown in Table A1, when $\lambda =1$, $r=0.05$, and $b=0$, the $AR{L}_1$ values corresponding to $a=1.25$ and $a=1.5$ are 63.61 and 18.89, respectively. When the scale parameters and location parameters of the two distributions shift at the same time, the performance of the EWMA-Lepage monitoring scheme is still satisfactory. Moreover, when $r=0.05$, the effectiveness of the EWMA-Lepage monitoring scheme is better in monitoring small to moderate shifts. For example, as observed from Table A1, when $a=1$ and $b=0.25$, the values of $AR{L}_1$ corresponding to $r=0.05$, $r=0.1$, and $r=0.2$ are 177.91, 198.07, and 212.21, respectively. The monitoring performance of the EWMA-Lepage scheme for $\lambda =1.5$ and $\lambda =2$ is presented in Table A2 and Table A3.

3.2.2. Analysis Under Pattern 2: $S\sim Normal(\mu ,{\sigma }^2)$ and $T\sim Weibull({\alpha }_1,{\beta }_1)$

In this subsection, we consider the case where S follows a normal distribution and T follows a Weibull distribution. For T, the shape parameter ${\beta }_1$ is fixed, while the scale parameter ${\alpha }_1$ is varied, and we set ${\alpha }_1=1,1.5,2$. The simulation results are presented in Table A4 and Table A6.

The performance of the EWMA-Lepage monitoring scheme is similar to that of the first pattern of distributions, namely, the case where S follows a normal distribution and T follows an exponential distribution, as described in Section 3.2.1. When the scale parameters do not shift, i.e., $a=1$, ${\alpha }_1=1$, and ${\beta }_1=1$, the EWMA-Lepage monitoring scheme performs well in detecting shifts in the location parameter. As shown in Table A4, when $r=0.05$, the $AR{L}_1$ values corresponding to $b=0.25$, $b=0.5$, and $b=0.75$ are 180.58, 33.27, and 10.49, respectively. When the location parameter remains unchanged, i.e., $b=0$, and the scale parameters a and ${\alpha }_1$ shift, the EWMA-Lepage scheme can also effectively monitor the shifts in the scale parameters. For example, as shown in Table A4 and Table A5, when $r=0.05$, $b=0$, and ${\alpha }_1=1$, the $AR{L}_1$ values corresponding to $a=1.25$ and $a=1.5$ are 64.99 and 18.73, respectively. When $r=0.05$, $b=0$, and ${\alpha }_1=1.5$, the $AR{L}_1$ values corresponding to $a=1.25$ and $a=1.5$ are 25.77 and 14.24, respectively. When both the scale and location parameters shift simultaneously, the monitoring performance of the scheme is also satisfactory. Moreover, compared to the smoothing parameters $r=0.1$, $0.2$, $0.3$, and $0.5$, $r=0.05$ still performs better in detecting small to moderate shifts.

3.2.3. Analysis Under Pattern 3: $S\sim Gamma(\alpha ,\beta )$ and $T\sim Exp\left(\lambda \right)$

Let $S\sim Gamma(\alpha ,\beta )$ and $T\sim Exp\left(\lambda \right)$, where $\alpha$ and $\beta$ denote the shape and scale parameters of the Gamma distribution, respectively. In this paper, we consider $\alpha =1,1.25,1.5,1.75,2,2.5,3$; $\beta =1,1.25,1.3,1.5$; and $\lambda =1,1.5,2$. We set the IC state to be $\alpha =\beta =1$. The performance of the EWMA–Lepage monitoring scheme is shown in Figure A1, Figure A2 and Figure A3. The effectiveness of the EWMA-Lepage monitoring scheme is influenced by variations in the shape and/or scale parameters of the gamma distribution, while the parameters of the exponential distribution are unchanged.

From Figure A1, Figure A2 and Figure A3, it can be observed that when $\beta$ is fixed, the logarithm of ARL₁ decreases as $\alpha$ increases. It can also be observed that larger smoothing parameters are suitable for detecting large shifts, while smaller smoothing parameters are suitable for detecting small shifts. Therefore, the EWMA-Lepage monitoring scheme is most effective for detecting small shifts when $r=0.05$, moderate shifts when $r=0.1$ or $0.2$, and large shifts when $r=0.3$ or $0.5$.

3.2.4. Analysis Under Pattern 4: $S\sim Gamma(\alpha ,\beta )$ and $T\sim Weibull({\alpha }_1,{\beta }_1)$

In this section, we set ${\alpha }_1=1,1.5,2$ and ${\beta }_1=1$. The monitoring performance of the scheme is shown in Figure A4, Figure A5 and Figure A6. When the parameters of the Weibull distribution remain unchanged, the performance of the EWMA-Lepage monitoring scheme is affected when either or both of the shape and scale parameters of the gamma distribution shift. When $\beta$ is fixed, as $\alpha$ increases, the logarithm of ARL₁ decreases, and the detection scheme can promptly monitor shifts caused by special causes. Consistent with the previous findings, we recommend using a small smoothing parameter to monitor small shifts.

To summarize, Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6 and Figure A1, Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6 show that the proposed EWMA-Lepage monitoring scheme has good performance for any shift of location, shape, or scale parameters, except when shifts in location parameters are small or negligible. Therefore, the proposed EWMA-Lepage scheme is effective in detecting shifts in both the location and scale parameters.

3.2.5. Overall Performance of the EWMA-Lepage Scheme

Since the actual shift size may not be known in practice, a monitoring scheme that performs consistently across different shifts regardless of specific shift information is preferred. In this paper, we simultaneously monitor two sample parameters. Therefore, according to Zhang et al. [40], we further use the following two performance indicators to compare the efficiency of different schemes.

• Expected $ARL$$\left(EARL\right)$. This simplified indicator is used to measure the overall performance of the monitoring scheme, which is called the expected average running length and is determined by:

  $$EARL={\displaystyle \frac{1}{pq}}{\sum }_{i=1}^p{\sum }_{j=1}^{q}ARL\left({\xi }_{1i},{\xi }_{2j}\right)$$

  where p and q are the numbers of T shifts and S shifts, respectively (such as, $p=28,q=3$ in Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6), and $ARL({\xi }_{1i},{\xi }_{2j})$ is the $ARL$ value of the shifts’ combination of $({\xi }_{1i},{\xi }_{2j})$. The scheme has better performance with small $EARL$.
• Relative $ARL$$\left(RARL\right)$. $RARL$ is one of the indicators used to evaluate the comprehensive performance of the scheme relative to the benchmark. It can be concluded that a $RARL$ value greater than 1 indicates inferior overall performance of the proposed scheme compared to the benchmark scheme and the expression of $RARL$ is as follows:

  $$RARL={\displaystyle \frac{1}{pq}}{\sum }_{i=1}^p{\sum }_{j=1}^q{\displaystyle \frac{ARL\left({\xi }_{1i},{\xi }_{2j}\right)}{AR{L}_{\mathrm{bmk}}\left({\xi }_{1i},{\xi }_{2j}\right)}}$$
  In this study, $AR{L}_{bmk}({\xi }_{1i},{\xi }_{2j})$ is the $ARL$ value of the shift combination of $({\xi }_{1i},{\xi }_{2j})$ when $r=0.05$.

In

Table 3. $EARL$ and $RARL$ values of the EWMA-Lepage scheme under four distribution patterns. ($m=100$, $n=5$, $AR{L}_0=370$).

EARL
	$r=0.05$	$r=0.1$	$r=0.2$	$r=0.3$	$r=0.5$
Pattern 1	18.86	20.24	22.88	24.93	28.29
Pattern 2	15.40	15.81	17.04	18.12	20.23
Pattern 3	22.85	24.75	28.49	30.88	35.33
Pattern 4	18.35	18.94	20.73	22.12	24.45
RARL
	$r=0.05$	$r=0.1$	$r=0.2$	$r=0.3$	$r=0.5$
Pattern 1	1.00	1.01	1.06	1.15	1.34
Pattern 2	1.00	0.96	1.01	1.05	1.20
Pattern 3	1.00	1.00	1.08	1.18	1.38
Pattern 4	1.00	0.97	1.00	1.06	1.22

, the values of $EARL$ and $RARL$ are presented under four different distribution combinations, reflecting the performance of the EWMA-Lepage monitoring scheme. Based on the $EARL$ values in Table 3, it can be observed that a smaller value of r corresponds to better performance of the EWMA-Lepage monitoring scheme. With a fixed value of r, Pattern 2 exhibits superior detection performance for the EWMA-Lepage monitoring scheme. According to the values of $RARL$, it can be observed that as r decreases, the overall performance of the EWMA-Lepage monitoring scheme improves.

4. Case Study

In this section, we will use the real online comments of Ctrip (https://www.ctrip.com) to demonstrate the implementation process of the EWMA-Lepage scheme. Ctrip plays a vital role in China’s online tourism service market, continues to be widely welcomed, and has been rated as China’s leading tourism group for four consecutive years. In the highly competitive online tourism industry, Ctrip places significant emphasis on monitoring and analyzing customer feedback to maintain its market position. Based on Chen et al. [44], findings suggest that issues related to hotel services and air ticket bookings represent key areas requiring improvement in Ctrip’s customer service. Therefore, this case focuses on the detection of service quality anomalies in chain hotels.

The data in this section are from Table E9 in the paper by Zhang et al. [40]. The negative evaluations in the first 4 months were included in the reference sample with a sample size of 50 (i.e., m = 50), and the remaining negative evaluations were used as test samples with a sample size of 5 (i.e., n = 5). The 20 test samples are shown in

Table 4. Values of S and T used as test samples in the EWMA-Lepage scheme for hotel service monitoring.

No.	S	T	No.	S	T	No.	S	T
1	8.78	3	8	14.16	3	15	5.15	0.2
	6.72	1		9.04	4		3.3	0.2
	3.95	1		8.39	3		4.16	0.2
	2.42	2		3.78	1		4.53	1
	14.03	0.33		6.71	3		3.91	2
2	11.36	0.33	9	4.4	3	16	7.35	1
	2.82	0.33		4.96	1		16.62	1
	15.39	4		4.38	1		2.61	0.5
	10.23	3		3.52	3		5.23	0.5
	5.34	5		11.5	2		2.46	3
3	4.6	1	10	5.26	2	17	9.19	2
	5.42	4		4.97	4		9.68	1
	11.7	2		6.13	3		7.22	1
	3.81	1		4.28	2		12.5	0.5
	8.38	1		3.41	1		5.49	2
4	7.52	2	11	2.84	1	18	7.87	3
	6.07	3		3.57	0.33		5.11	1
	7.57	2		4.06	0.33		9.52	4
	6.27	1		3.57	1		6.49	1
	7.06	4		6.49	1		4.3	3
5	6.85	2	12	3.39	6	19	10.16	0.33
	7.21	1		2.96	1		5.32	0.33
	3.57	4		6.42	2		4.3	0.33
	8.69	1		4.02	6		8.55	1
	7.19	1		5.85	1		6.8	0.33
6	5.65	2	13	4.16	4	20	6.93	0.5
	6.93	4		5.6	4		13.34	0.5
	5.86	1		1.44	1		12.28	0.5
	26.7	1		6.15	3		14.8	0.5
	4.66	1		11.97	2		18.42	0.5
7	18.41	1	14	8.1	2
	9.12	3		7.62	2
	4.95	1		4.39	2
	7.07	2		5.37	1
	3.5	2		5.53	4

. If there are multiple negative comments within a single day, the time interval is calculated as 1 divided by the number of negative comments on that day. In addition, we used Minitab (version 17) software to test the goodness of fit of the reference samples of S and T at the level of 5 percent significance. It can be found that the observed values of S can be fitted to multiple distributions. To utilize the nonparametric EWMA-Lepage scheme, we set $AR{L}_0=370,r=0.05$, and the control limit of the scheme is set to 2.7495.

As shown in Figure 2, the EWMA-Lepage scheme detects two OOC signals at the 19th and 20th test samples, indicating that the scheme is effective. The two OOC signals may be attributed to shifts in S and T, with S being the dominant contributing factor. According to Zhang et al.’s [40] study, from the original manual inspection comments, we found that many customers complained about noise pollution, which is due to the problem of illegal building at night. In this case, the hotel should contact the construction manager as soon as possible and take some measures to alleviate customer complaints.

5. Conclusions

Negative comments offer valuable insights into hidden quality issues in online services. Detecting and addressing their abnormal shifts is crucial for businesses. This study proposes a nonparametric EWMA-Lepage scheme to jointly monitor the emotional score S and the time interval T from sequentially collected negative comments, thereby detecting shifts in both location and scale parameters, and this scheme combines the EWMA-Lepage statistics derived from S and T through a max-type combination function, allowing for the simultaneous detection of upward and downward shifts in the monitored parameters. The IC and OOC performances of the EWMA-Lepage scheme are evaluated by various performance metrics including the $ARL$, $SDRL$, some percentiles, $EARL$, and $RARL$. In the IC state, we find that the EWMA-Lepage scheme proposed in this paper has a combination of many specifications $(m,n,r)$, all of which have satisfactory performance. In addition, in the OOC scenario, the proposed distribution-free EWMA-Lepage scheme demonstrates that smaller smoothing parameters are suitable for detecting small shifts, while larger smoothing parameters are more effective for detecting large shifts. Furthermore, the scheme shows good performance in detecting quality or service issues during the monitoring of online customer feedback.

While the proposed distribution-free EWMA-Lepage scheme demonstrates effective performance in monitoring bivariate processes of sentiment intensity and time-between-events for online reviews, several promising avenues for future research remain. (1) Future research should explore monitoring schemes that account for potential dependence between sentiment score and time interval, possibly using copula models or multivariate time series approaches. (2) This study focuses on a single platform and specific categories for in-depth analysis; extending the framework to multiple platforms, diverse categories, higher-dimensional monitoring, and cross-validation is a key direction for future research. (3) Future studies could integrate more advanced natural language processing techniques, such as deep learning models fine-tuned on domain-specific review data, to enhance the accuracy and robustness of sentiment scoring and improve overall monitoring performance.