Nonparametric EWMA-Type Control Charts for Monitoring Industrial Processes: An Overview

In the present paper we provide an up-to-date overview of nonparametric Exponentially Weighted Moving Average (EWMA) control charts. Due to their nonparametric nature, such memory-type schemes are proved to be very useful for monitoring industrial processes, where the output cannot match to a particular probability distribution. Several fundamental contributions on the topic are mentioned, while recent advances are also presented in some detail. In addition, some practical applications of the nonparametric EWMA-type control charts are highlighted, in order to emphasize their crucial role in the contemporary online statistical process control. KeywordsDistribution-free statistical methods, Rank-based procedures, Nonparametric statistical process control, Exponentially weighted moving average (EWMA) control charts, Sign statistics.


Introduction
Statistical process control is implemented in industrial procedures and aims at improving their quality by monitoring the corresponding production process. Among others, monitoring schemes have been proved to be very useful statistical methods for supervising a measured characteristic of the product and tracking down plausible changes in the underlying distribution of the process. The most popular and easy-to-use monitoring schemes, which have been introduced and well-studied in the literature, are the so-called Shewhart-type, Cumulative Sum (CUSUM, hereafter) and Exponentially Weighted Moving Average (EWMA, hereafter) monitoring schemes. For a thorough consideration of the basic principles of the topic, the excellent monographs of Montgomery (2009) and Qiu. (2014) or the works of Woodall (2000) and Woodall and Montgomery (1999) are strongly recommended.
A broad variety of monitoring schemes have been already established under the presumption that the production data match adequately a specific probability distribution. Nevertheless, this demand is not often fulfilled and consequently it is evident that the distribution-based monitoring schemes do not constantly provide a reliable framework. To overcome this problem and at the same time maintain the common framework of a control chart, a broad variety of nonparametric schemes have been introduced in the literature. It is forthright that nonparametric (or distribution-free) control charts do not rest with the process distribution and therefore there is no need to assume one (or even worse to be obliged to know it). The general framework of the so-called Nonparametric Statistical Process Control and some fundamental perspectives of it could be found in the works of Chakraborti and Graham (2019a), Li (2011a, 2011b) or Qiu (2018).
One of the classes of distribution-free schemes for supervising univariate processes, is the family of Shewhart-type nonparametric control charts (SNCC class, hereafter). The schemes, which are included in the SNCC class, utilize an appropriate nonparametric test statistic, whose values are calculated by the aid of individual observations or random groups collected from the process. These observations are depicted and compared to the control limits of the corresponding chart, in order to draw a conclusion about the status of the production. The SNCC class seems to include the easiestto-use control charts in comparison with the other families of distribution-free schemes. Some pioneer works on the SNCC class are presented by the review paper of Chakraborti et al. (2001), while recent advances on the topic are discussed by Chakraborti and Graham (2019b) or Koutras and Triantafyllou (2020).
A second family of nonparametric schemes for monitoring univariate processes includes the socalled CUSUM nonparametric control charts (CNCC class, hereafter). Each member of this class calls for an appropriate nonparametric test statistic and afterwards the CUSUM procedure is constructed via the conventional approach. The CNCC class outperforms the SNCC family for tracking down slight or middling changes of either the location or the scale parameter of the underlying process. For multivariate CUSUM nonparametric control charts, the works provided by Qiu and Hawkins (2001), Qiu (208) or Xue and Qiu (2020) are highly recommended.
The third family of nonparametric monitoring schemes contains memory-type statistical tools, known as nonparametric EWMA control charts (EWMANCC class, hereafter). These charts utilize an appropriate nonparametric testing procedure in order to build the corresponding EWMA statistic, whose values are afterwards plotted and monitored by the aid of the respective control limits. It is evident that the EWMANCC class is supposed to be superior to the aforementioned SNCC class for tracking down small changes of the underlying distribution process. Throughout the lines of the present manuscript, the EWMANCC class is under investigation. Two main subgroups of the EWMANCC class are studied in detail. The first one includes the EWMA-type schemes based on sign statistics, while the second subgroup contains the corresponding rank-based ones. Our study is devoted only to these subgroups due to their enormous growth during the last decade. It is noticeable that over 20 different nonparametric control charts, which belong to the aforementioned subgroups, have been introduced in the last two years. However, it is important to mention that a variety of EWMA-type nonparametric monitoring schemes based on alternative approaches have been also established in the literature (see, e.g. Zou and Tsung, 2010;Graham et al., 2012Graham et al., , 2016;) Qiu et al., 2018;You and Qiu, 2020).
The review article is worked out as follows: Section 2 offers some general notations and the framework for constructing an EWMA control chart. A review of univariate nonparametric EWMAtype control schemes which rely on sign statistics is discussed in Section 3. Section 4 offers a detailed presentation of EWMA -type rank-based monitoring schemes, while Section 5 puts forth the general outline of the present manuscript, whilst some hints for further research on the topic are also dropped.

General Notations and EWMA Framework
Generally speaking, the EWMA framework relies on an exponentially weighted moving average of the ongoing and historical data. In simple words, if , 1, 2,... t Yt denotes the sequentially observed data, then the monitoring EWMA-type statistic Zt can be expressed as 1 (1 ) , 1, 2,...
where the starting value is often determined to be equal to the corresponding target value μ0 of the underlying process (see, e.g. Lucas and Saccucci, 1990). Note that the control limits of the resulting scheme are defined via the standard deviation Z  of the EWMA-type statistic appeared in (1) and are usually chosen as 0 , where the design parameter L is appropriately determined such that the desired performance level is reached. For some fundamental advances on the topic of EWMA-type control charts, the works provided by Hunter (1986) or Roberts (1959) are recommended.
When a process is under investigation, a common way to monitor its performance is to utilize the so-called sign statistic for providing some evidence about the behavior of a specific measurable characteristic of the product. If we denote by τ the target value of the underlying quality characteristic, the sign statistic is simply an indicator variable, which provides the information whether each observed value of the underlying characteristic is above or below the target value. However, plenty alterations of the traditional sign statistic have been already studied and the corresponding control charts have been studied in detail.
We next provide a detailed review of nonparametric univariate monitoring schemes based on sign statistics, namely of these members of the EWMANCC class which employ the well-known sign statistic or its modifications. The set-up of presentation of the included articles is determined to be chronological for the purpose of setting up a comprehensible and well-structured flow. where sign(a) takes on the value 1 (-1) if the quantity a is positive (negative). Amin and Searcy (1991) considered the following EWMA-type monitoring statistic where 1 g t tj j SR U    , denotes the signed-rank statistic for the t-th random sample of magnitude g.
The proposed two-sided EWMA-type scheme generates an out-of-control indication if the monitoring statistic Zt violates the upper or the lower control limit. It should be mentioned that in order for the corresponding chart to be capable to generate an alarm due to an observed random sample, the maximum value for the corresponding upper control limit is determined as ( 1) / 2 gg   which actually refers to maximum value that may be calculated for the variable Zt given that Z0 equals to zero. Amin and Searcy (1991) also investigated how the presence of autocorrelation between the observed values affects the behavior of the resulting scheme. Considering the first order autoregressive time series model of Box and Jenkins (1976), Amin and Searcy (1991) concluded that the in-control ARL of the respective scheme change roughly in a linear fashion as the autoregressive coefficient of the aforementioned model changes, in contrast with the exponential change of the corresponding ARL of the common EWMA procedure. For a thorough discussion about the above-mentioned issue, the interested reader is referred to Hunter (1990), MacGregor and Harris (1990) or Baxley (1990).
As Amin and Searcy (1991) argued, the proposed nonparametric EWMA monitoring scheme seems to be affected slightly by the specific choice of the corresponding weighting parameter. Amin and Searcy carried out a numerical experimentation based on different distributional models, such as the Normal distribution, the Uniform distribution, the Double Exponential distribution, the Gamma distribution and the Cauchy distribution. According to their simulation results, the proposed nonparametric EWMA monitoring scheme performs satisfactorily for non-normal or heavy-tailed distributions.
In addition, Graham et al. (2011a) studied the nonparametric EWMA control chart established by Amin and Searcy (1991) in some detail. Implementing a suitable Markov chain technique, the above-mentioned authors determined the distribution of the run length for the resulting scheme given that the process has not shifted, while practical guidance and suggestions for picking up its design parameters are also documented. Based on several quantitative measurements, just like the common ARL, the MRL or the SDRL, authors appraised the in-and out-of-control attribution of the nonparametric EWMA control scheme established by Amin and Searcy (1991). Their simulation procedure relied on random samples generated by several distributions, such as Standard Normal, scaled Student's t, Laplace or contaminated Normal distribution.
The control limits of the above-mentioned chart are given as ( 1)(2 1) 62 . The nonparametric signed rank statistic under the ranked set sampling framework is given by (see, e.g. Kim and Kim (1996) The in-control expected value and the variance of the test statistic defined in Eq. (5) are given by 2 0  Abid et al. (2017) proposed a distribution-free EWMA scheme, which takes into account the following plotted statistic: where i RSS SR is defined in Eq. (5) and λ is the smoothing parameter of the proposed EWMA scheme ( 01   ). The abovementioned statistic is plotted against the control limits: and the proposed distribution-free scheme generates an out-of-control indication whenever . It is forthright that the design parameter L of the resulting chart is appropriately determined so that a required degree of attribution is achieved.
On the other hand, Perdikis et al. (2020b) studied a altered distribution-free EWMA-type control scheme which utilizes the Wilcoxon signed rank statistic. In order to determine both the in-and out-of-control ARL of the resulting chart, Perdikis et al. (2020b) utilized a continuous transformation of the Wilcoxon signed rank statistic, combined with the common Markov chain technique (see, e.g. Brook and Evans, 1972). In their framework, the structure of an EWMA scheme which could be proved sensitive in tracking down increases in the quality measurement of interest is only considered.
Based on Eq.
(2), one may readily deduce that the so-called Wilcoxon signed-rank statistic coincides to the sum of the signed ranks defined on   ( 1) / 2, ( 1) / 2 2,..., ( 1) / 2 2, ( 1) / 2 where t SR  corresponds to the sum of the positive ranks. Note that following a generating function approach proposed by McCornack (1965), the probability mass function of variable t SR  could be readily delivered. Implementing a parallel argumentation as the one proposed by Wu et al. (2020), Perdikis et al. (2020b) suggested to transform the statistic t SR into a new continuous one denoted by * t SR , whose probability density function can be expressed by the aid of the corresponding function of variable t SR  . Consequently, the monitoring statistic of the proposed continuousified one-sided EWMA-type control chart is given by: ), with 0 where λ corresponds to the smoothing parameter. The upper-sided EWMA-type monitoring scheme proposed by Perdikis et al. (2020b) generates an out-of-control alarm if the statistic defined in Eq.
(12) exceeds the corresponding upper control limit, which is given as: while p0 corresponds to the in-control probability of recording an observation greater than the target value and σ denotes the "continuousify" parameter. Perdikis et al. (2020b) investigated the efficiency of their proposal by utilizing plenty kernels apart from the Gaussian and they deduced that no important differences are present. Note that their study was mostly aimed at offering a powerful approach for examining the chart's run length attributes rather than confirming its supremacy against other competitors.
In addition, Perdikis et al. (2020a) established a nonparametric EWMA-type control chart based on the recursive formula introduced by Rakitzis et al. (2015), where the Wilcoxon signed rank statistic defined earlier is now utilized. An appropriate discrete Markov chain technique is implemented for determining the ARL of both in-and out-of-control cases. Perdikis et al. (2020a) considered the monitoring statistic Yt which is obtained by the aid of the following formula: where , ty  are two fixed positive integer-valued parameters, the initial values R0 and Y0 are equal to zero and t SR is given by Eq. (11).
Given that the distribution of test statistic t SR under the hypothesis that the process has not shifted, is proved to be symmetric around 0, the corresponding limits and the center line of the proposed scheme coincide to LCL = −K, CL = 0 and UCL = K respectively, where K takes on values from the set {2, 4,..., ( 1) / 2} gg . It is straightforward that the proposed nonparametric EWMA scheme proposed by Perdikis et al. (2020a) produces an out-of-control signal if In order to optimize the performance of their proposed monitoring scheme, Perdikis et al. (2020a) utilized an appropriate algorithmic procedure for finding out the optimal combination of the design parameters, which results in the minimum respective out-of-control ARL.

Nonparametric EWMA Sign Schemes Based on Sequential Records
Yang et al. (2011) established a nonparametric EWMA control scheme for variables data to supervise the dispersion from the process target, without presuming a specific distributional model.
Based on independent random samples of magnitude n, say 12 , ,..., n X X X , which are drawn from the process, the variables Yj and Ij are defined as:  . Note that the design parameters are determined appropriately for accomplishing a pre-specified performance level. Note that Yang et al. (2011) also applied the so-called arcsine transformation (see, e.g. Mosteller and Youtz, 1961) and constructed a revised version of their proposed control chart. A discussion referring to the abovementioned work is provided by Abbasi (2012).
Following their proposal, Aslam et al. (2014) modified the distribution-free EWMA scheme which relies on the sign statistic defined earlier, by implementing the so-called repetitive sampling procedure. Generally speaking, the aforementioned sampling scheme, which has been established by Sherman (1965), seems to be easy-to-use and more capable than the corresponding single or double sampling procedures (see, e.g. Balamurali and Jun, 2006).
Since the repetitive sampling scheme is activated, the conclusion about the status of the process relies on a single sample only if it is conspicuous, but resampling will be considered if it is questionable. In other words, the nonparametric EWMA monitoring scheme proposed by Aslam et al. (2014) is constructed by the aid of double control limits, namely two different lower and two separate upper control limits are calculated and considered simultaneously for monitoring the underlying process. More precisely, the distribution-free control chart considered by Aslam et al. (2014) generates an alarm whenever one of the following occurs: then the process is characterized as in-control. Of course, the case where the monitoring statistic is located between the two upper control limits UCL1, UCL2 or between the two lower control limits LCL1, LCL2, is not covered by the above-mentioned conditions stated in (19) and (21). In such case, the proposed procedure calls for resampling, namely a different random sample of the same magnitude is collected from the process and the EWMA sign statistic is computed once again by the aid of formula given above. For illustration purposes, Aslam et al. (2014) applied the proposed EWMA nonparametric control chart for real data concerning the fill volume of soft-drink beverage bottles (see, e.g. Montgomery, 2009).
Moreover, Abid et al. (2016) introduced a distribution-free EWMA scheme which utilizes the sign test statistic and implements the so-called ranked set sampling, which seems to be useful especially when measurements are ruinous or costly. The proposed distribution-free scheme is an attractive second choice instead of the common distribution-free sign scheme mentioned earlier. Under the ranked set sampling, a single group of magnitude n is collected randomly from a particular pool and afterwards, authorities order the units. The observation with the lowest rank is detected and forwarded for quantification. The rest n − 1 units are put back to the pool and a different group of same size is collected. Again, the authorities rank the units and the second lowest is detected, quantified and the rest n − 1 units are put back once again. The process is done over until n ordered units are measured, say (1)1 (2)1 ,..., n X X X , which corresponds to the random sample of the first cycle. The same procedure is realized for m repetitions and an ordered group of magnitude r=nm is obtained. Some advances on ranked-set-sampling-based process control, are provided by Abujiya and Muttlak (2004), Muttlak and Al-Sabah (2003), Al-Omari and Haq (2012), Mehmood et al. (2013) or Haq et al. (2015).
The most handy conformation of the ranked set sampling sign test statistic can be viewed as (see, e.g. Hettmansperger, 1995).
where the above-mentioned variable I() can be expressed as: Note that the test statistic defined in Eq. (18) can be viewed as a sum of n () , 1, 2,...,


. It goes without saying that the design parameter k of the resulting chart corresponds to the distance between the lower and upper limit and is appropriately determined for reaching a desired level of attribution.
Given that the distribution of test statistic t SN under the hypothesis that the process has not shifted, is symmetric around 0, the control limits and the center line of the proposed scheme are equal to LCL = −K, CL = 0 and UCL = K respectively, where K takes on integer even values equal to or greater than 2 and less than or equal to the even n, or integer odd values greater than or equal to 1 and less than or equal to the odd n. It is straightforward that the proposed distribution-free EWMA scheme proposed by Castagliola et al. (2019) Graham et al. (2011b) established a distribution-free EWMA sign scheme for independent and identically distributed individual observations. A Markov chain technique is implemented for deducing the distribution of run length for the resulting scheme, while several performance characteristics are studied in order to shed light on its in-and out-of-control behavior. Graham et al. (2011b) considered the well-known sign statistic SNi defined as:

A Distribution-Free Sign Scheme Based on Individual Measurements
where θ0 corresponds to the target value. In their framework, the monitoring statistic Zi of the proposed EWMA scheme is deduced by successively accumulating the sign statistics , 1, 2,...
whereas λ denotes its smoothing parameter ( 01   ). Note that Graham et al. (2011b) considered only the median value among other available percentiles and consequently throughout the lines of their study, the parameter θ0 expresses the in-control process median. Following their choice, the control limits of the proposed EWMA nonparametric monitoring scheme are given by: , 22 Graham et al. (2011b) presented several combinations of the design parameters λ and L, which result in a required degree of in-control attribution. Based on the above-mentioned results, the practitioner has at his/her disposal a useful guidance for implementing the proposed control chart.  Graham et al. (2011b) appraised the performance of the proposed EWMA monitoring scheme by considering different distributional models. Their numerical experimentation included symmetric and asymmetric distributions, light-and heavy-tailed distributions, while the effect of outliers was also studied. For slighter changes, the common EWMA scheme tracks down changes faster than the nonparametric one if the normal distribution is assumed. Conversely, the proposed nonparametric monitoring scheme is more capable versus the ordinary EWMA scheme under the more highly asymmetric and the more heavy-tailed distributions, such as the Laplace or Gamma case. Lu (2015) established a nonparametric generally weighted moving average (GWMA) sign scheme, which aims at improving the detection ability in slight changes of the process. The proposed scheme adopts the framework of the distribution-free scheme introduced by Yang et al. (2011) and extends the idea to the so-called generally weighted moving average scheme (see, e.g. Sheu and Lin, 2003).

An Extended Distribution-Free EWMA Sign Scheme
Let X be the measurement of concern with target value θ, while Y = X-θ expresses the corresponding distance of X from the target. In this framework, Lu (2015) defined the process proportion p as the probability of observing positive Y's and consequently the process is declared to be in-control (out-of-control) if p equals to (is unequal with) 0.5. Based on independent random samples of magnitude n, say 12 , ,..., , 1, 2,...
, which are drawn from the process at time t, the variables Yit and I are defined as: respectively. If Mt corresponds to the total number of positive deviations Yit's, then it is straightforward that the random sum , follows the Binomial distribution with parameters n and p = 0.5, under the presumption that the process is in-control. Lu (2015) introduced the nonparametric GWMA monitoring scheme, whose plotted statistic is given by: where Mt corresponds to the observation from the production at time t.
It goes without saying that the resulting nonparametric sign chart generates an alarm whenever or tt G UCL G LCL  . Note that the design parameters are determined appropriately in order to reach the pre-determined degree of attribution. It is noteworthy that the nonparametric chart proposed by Lu (2015) is a generalization of the respective EWMA monitoring scheme proposed by Yang et al. (2011). Indeed, if we consider the special case w = 1-λ, then Eq. (34) reduces to the sign control chart of the above-mentioned authors. Due to some simulation results, Lu (2015) concluded that the proposed GWMA monitoring scheme is proved to be more efficient for tracking down small process changes versus other EWMA schemes considered, while its implementation is strongly recommended under large values of the parameter w and a. Riaz (2015) proposed a distribution-free scheme which relies on moving sign statistics in an EWMA framework. Let X be the measurement of our concern with target value μ, while Y = X-μ expresses the corresponding distance of X from the objective. In a similar framework as the one applied by Lu (2015), the probability p of observing positive Y's could be utilized for monitoring the process location parameter. For an in-control process, probability p is presumed to be equal to p0 , while in case of an out-of-control change the proportion is denoted by p1.

A Distribution-Free Sensitive EWMA Sign Scheme on A Moving Pattern
Based on independent random samples of magnitude n, say 12 , ,..., , 1, 2,... (1 ) , where λ corresponds to the smoothing constant (0<λ<1), while the statistic Qt is viewed as the average value of Given that the initial value of the above-mentioned EWMA scheme is given by 00 W np  , the corresponding mean and variance can be expressed by the aid of the following: The EWMA-type nonparametric monitoring scheme can now be constructed by the aid of the following control limits: where L corresponds to the multiplier utilized for mending the wrong alarms. The resulting twosided control scheme generates an alarm if the monitoring statistic Wt defined in Eq. (36) is larger than UCL or smaller than LCL. Riaz (2015) appraised the attribution of the resulting control scheme by the aid of its run length. More precisely, the ARL, the SDRL, the RARL, the EQL and the PCI were considered in order to examine the in-and out-of-control behavior of the proposed nonparametric EWMA scheme.
According to the comparative results provided by Riaz (2015), the proposed scheme seems to outperform other competitive CUSUM and EWMA charts, while its run length curve indicated the higher probability of detection compared to other schemes. Riaz (2015) argued that the efficiency of the resulting scheme could be even more strengthened by swelling the sequence of moving sign statistics or applying runs rules. Riaz et al. (2019) established a distribution-free arcsine EWMA sign control scheme by implementing the so-called sequential sampling scheme (see, e.g. Wald, 1943). The sequential sampling scheme is a sampling plan in which an undefined amount of groups are examined successively, accumulating the results, until a conclusion is drawn. Note that in the framework of sequential sampling procedure, the sample magnitude is not pre-specified.

An Enhanced Distribution-Free EWMA Sign Scheme Utilizing Sequential Mechanism
The construction of the control chart established by Riaz et al. (2019) calls for the determination of two pairs of control limits LCL1, UCL1, LCL2, UCL2 according to the following steps.
Step 1. A random group of magnitude n is collected and the following EWMA-type monitoring statistic is calculated: where T denotes the arcsine transformation, namely 1 sin ( ) Tp   while p corresponds to the proportion of positive deviations.
Step 2. The control limits of the resulting scheme are now calculated by the aid of the following equations then the process is characterized as in-control. Of course, the case where the monitoring statistic is located between the two upper control limits UCL1, UCL2 or between the two lower control limits LCL1, LCL2, is not covered by the above-mentioned conditions stated in (42) and (43). In such case, the proposed procedure calls for resampling, namely a separate random sample of the same magnitude is collected from the process and the EWMA-type statistic is calculated once again by the aid of formula given above. Riaz et al. (2019) carried out a numerical comparative analysis among control charts which utilize single, double, repetitive and sequential sampling schemes. For illustration purposes, a real-data problem referring to smartphone accelerometer data is presented in some detail (see also Riaz et al., 2017).
The attribution of the proposed monitoring scheme is appraised and contrasted to its counterparts, by the aid of the corresponding run length properties such as the ARL, the MRL or the SDRL. According to the numerical results provided by Riaz et al. (2019), the proposed distribution-free EWMA-type monitoring scheme is proved to be efficient in tracking down slight changes of the underlying process distribution. Tang et al. (2019b) introduced a nonparametric adaptive EWMA control scheme for count data. Generally speaking, the adaptive EWMA schemes combine the well-known Shewhart-type and the traditional EWMA framework in a smooth way (see, e.g. Capizzi and Masarotto, 2003;Tang et al., 2019a;Aly et al., 2017). Their proposed chart is suitable for monitoring count data and consequently the corresponding plotting statistic takes on only integer values, while the respective weights should be positive integer-valued. A performance analysis based on exact run length properties is carried out by implementing an appropriate discrete-time Markov chain technique.

A Distribution-Free Adaptive EWMA Sign Control Chart
The nonparametric control scheme proposed by Tang et al. (2019b) utilizes the following adaptive EWMA scheme for count data.
Note that the term Yt can be viewed as the rounded towards zero integer part of the quotient of the Euclidean division

A Nonparametric Synthetic EWMA Control Chart Based on Signs
Haq (2019) proposed a synthetic EWMA sign scheme by integrating the existing and well-known EWMA sign scheme with the conventional run length scheme, which was first introduced by Bourke (1991) based on data on individual items for controlling a specific characteristic. The socalled conforming run length chart has been already mixed with Shewhart-type, CUSUM-type and EWMA-type monitoring schemes resulting in several synthetic control charts (see, e.g. Wu and Spedding, 2000;Scariano and Calzada, 2009 The step-by-step testing procedure of the resulting monitoring scheme could now be boiled down as follows: If at time t the observed value of the EWMA-type statistic Ht based on the corresponding random sample of magnitude n lies between the control limits defined in Eq. (48), then the particular sample is declared to be a conforming sample and another random sample of same magnitude should be drawn. However, in different case the monitoring procedure goes on to the next stage. (ii) The number of groups between the ongoing and the last nonconforming group is counted, namely the observed CRL sample of the corresponding control chaschemert is calculated.
Once the value CRL is equal to or less than L, then the proposed EWMA synthetic scheme generates an out-of-control alarm.
According to the run length performance comparisons provided by Haq (2019), it turns out that the resulting EWMA synthetic monitoring scheme is more sensitive than the EWMA sign scheme established by Yang et al. (2011). Ali et al. (2020) introduced a distribution-free EWMA control scheme based on sign statistics under a ranked set sampling scheme (see, e.g. McIntyre, 1952). Some general details about the specific sampling scheme are given in subsection 2.2 of the present manuscript.

A Nonparametric EWMA Sign Control Scheme Under the Ranked Set Sampling
where the quantity 2 0  reveals the advantages of ranked set sampling over standard random sampling and is provided by Hettmansperger (1995 respectively. It goes without saying that the resulting nonparametric EWMA scheme generates an out-of-control alarm if or qq REWMA UCL REWMA LCL  . Note that the design parameters are determined appropriately for reaching a required degree of attribution. According to some numerical results provided by the authors the in-control robustness and superiority of the resulting chart over its competitive ones in tracking down all types of changes in the process location is revealed. Raza et al. (2020) established a distribution-free double EWMA control chart based on signed-rank statistics. More specifically, the resulting monitoring chart is constructed by melting the Wilcoxon signed-rank (SR) statistic into the EWMA framework and then combining two EWMA statistics with the motivation of enhancing the sensitivity of the resulting chart. Let , 1, 2,..., it Xt  denote the i-th observed value of the t-th random sample for the quality characteristic of interest with target value θ0. In their framework, the singed rank statistic is defined as: 1 , 1, 2,...,

A Nonparametric Double EWMA Sign Control Chart
where, respectively. Note that parameter k simply expresses how many times the standard deviation of the monitoring statistic wriggles into the decision interval of the resulting chart.   Shafqat et al. (2020) introduced a linear prediction related double EWMA sign control scheme utilizing the repetitive sampling plan. The resulting control scheme is suitable for monitoring binomially distributed data and seems to be efficient in tracking down plausible small changes of the underlying process.

A Distribution-Free Repetitive Sampling DEWMA Scheme Based on Linear Prediction
Following a similar argumentation as the one provided by Brown and Meyer (1961), Shafqat et al. (2020) constructed a double EWMA control chart for delivering linear predictions of the form F  corresponds to the forecast in the period t while the coefficients ai and bi are given as: Note that Zi is the arcsine-transformed EWMA sign statistic mentioned earlier, while i Z corresponds to the monitoring statistic of the respective double EWMA control chart (see, e.g. Shamma and Shamma, 1992).
The asymptotic control limits of nonparametric linear prediction-based double EWMA control chart designed for repetitive sampling are determined by the aid of the following equations: 1 (1 4(1 ) 5(1 ) ) 2 (1 3(1 )) (1 3(1 )) () the control coefficients k1 and k2 take on values depending on the other design parameters of the scheme as well as its required in-or out-of-control level of attribution. The nonparametric chart proposed by Shafqat et al. (2020) Wu et al. (2020) established nonparametric control chart for joint monitoring the time interval T between successive occurrences of an event E and its magnitude X. The authors also established a "continuousify" technique combined with a common Markov chain technique in order to International Journal of Mathematical, Engineering and Management Sciences Vol. 6, No. 3, 708-751, 2021 https://doi.org/10.33889/IJMEMS.2021.6.3.044 729 investigate the run length distribution and its characteristics. It is evident that several schemes which are suitable for monitoring both the time interval T and the magnitude X have already been studied in the literature (see, e.g. Wu et al., 2009;Qu et al., 2018;Rahali et al., 2019).

A Nonparametric EWMA Control Scheme for Monitoring Time-Between-Events-And-Amplitude Data
Let D0=0, D1, D2, … be the dates of occurrence of a specific negative event E, while while the corresponding upper control limit is given by: For illustration purposes, Wu et al. (2020) studied a real data set concerning the time (in days) between fires in forests of a French region and their magnitudes.  Aslam et al. (2020) established a nonparametric sign control scheme which relies on the modified or controlled EWMA statistic. More precisely, the indicator Ij defined in Eq. (16) of the present manuscript is utilized, while the random sum 1 n j j MI    , which follows the binomial distribution with parameters n and p = 0.5, under the presumption that the process is in-control is also considered. Aslam et al. (2020) proposed a modified EWMA monitoring scheme by involving further controlled term in the EWMA sign statistic utilized by Yang et al. (2011). The proposed controlled modified EWMA-type sign statistic is expressed as:

A Nonparametric Sign Control Chart Based on A Controlled EWMA Statistic
Note that the proposed modified EWMA scheme defined in Eq. (60) reduces to the EWMA sign monitoring scheme of Yang et al. (2011) if we set k equal to zero.
The control limits of the two-sided control chart proposed by Aslam et al. (2020) are given below: It is forthright that the proposed distribution-free modified EWMA sign chart generates an out-ofcontrol alarm if The resulting scheme is contrasted to other distribution-free EWMA sign control schemes. According to the numerical results provided by the same authors, one may readily deduce that the resulting scheme is more sensitive than the competitive EWMA sign scheme in terms of run length characteristics. Moreover, Aslam et al. (2020) argued that the resulting control scheme is an attractive alternative for practitioners, especially when the aim is to track down slight to middling changes in the process. Alevizakos et al. (2020b) established a nonparametric double GWMA control scheme based on the signed-rank statistic for monitoring the location parameter of the underlying process. The run length distribution of the resulting chart is investigated in some detail, while plenty numerical calculations have been accomplished based on proper Monte Carlo simulation procedures. Let , 1, 2,..., it Xt  denote the i-th observed value of the t-th random sample for the quality characteristic of interest with target value θ0. In their framework, the signed rank statistic SR defined earlier is recalled (see, e.g. Eq. (52) of the present manuscript) in order to define the charting statistic of the proposed double GWMA scheme via the system of the following equations: ( 1)(2 1) ( 1)(2 1) , 66

A Nonparametric Double Generally Weighted Moving Average Signed-Rank Control Chart
where 2 1 t tj j Ww    , is the sum of squares of the weights and L corresponds to the width of control limits. It is evident that for large values of t, the asymptotic or steady-state control limits of the proposed monitoring scheme can be written as: ( 1)(2 1) ( 1)(2 1) , 66 where, It is forthright that the resulting nonparametric sign scheme generates an alarm if the charting statistic is located out of the region determined by the corresponding control limits.
According to the extensive numerical results provided by Alevizakos et al. (2020b), the proposed double GWMA monitoring scheme seems to be more efficient than its competitors for slight changes in the location parameter, while its attribution is similar to the other nonparametric charts for bigger changes. Alevizakos et al. (2020a) proposed a triple EWMA control scheme which utilizes the sign statistic for monitoring the location parameter of the underlying process. The run length-based attribution characteristics of the resulting scheme are studied in detail, while several numerical comparisons revealed its superiority against other competitors, especially for slight changes. The indicator defined in Eq. (16) of the present manuscript is utilized, while the random binomial variable 1 n t it i NI    , with parameters n and p = 0.5, under the presumption that the process is in-control is also considered. The charting statistic of the proposed triple EWMA scheme can be evaluated through the system of the following equation:

A Distribution-Free Triple EWMA Sign Control Scheme
where 01   is the smoothing parameter of the resulting chart while the initial values are set to be equal to n/2. The time-varying control limits of the two-sided monitoring scheme proposed by Alevizakos et al. (2020a) are given below: where the corresponding center line is given as /2 CL n  , while the in-control variance of the statistic Wt appeared in Eq. (66) can be computed by the aid of the following formula: ( 1)( 2) 4 ( 1) It is evident that for large values of t, the asymptotic or steady-state control limits of the proposed monitoring scheme can be written as: (2 ) (2 ) (2 ) 4 4 2 2 3 4 5 4 3 2 6(1 ) 12(1 ) 7(1 ) /2 (2 ) (2 ) (2 ) (2 ) 4 It is forthright that the proposed distribution-free scheme generates an out-of-control alarm if the charting statistic is located out of the region determined by the corresponding control limits. According to the extensive numerical results provided by Alevizakos et al. (2020b), the proposed double GWMA control scheme seems to be more efficient than its competitors for slight changes in the location parameter, while its attribution seems to be almost similar compared to the other nonparametric charts for bigger changes.

A Nonparametric EWMA Control Chart Based on Auxiliary Information
Haq (2020) introduced a nonparametric EWMA sign control chart by the aid of an auxiliaryinformation-based estimator of the process mean. The run length characteristics of the proposed scheme are investigated via an appropriate Markov chain approach. Let Y be the measurement of our concern with target value μΥ, while the probability of observing a Y-value greater than μΥ is denoted by p. Suppose next that there is an auxiliary characteristic X being related to the measurement Y. It is also presumed that the underlying joint probability distribution of (Y, X) is unascertained in contrast to the population parameters. Denoting by , , , X Y X Y     the population means and variances of X and Y, an unbiased estimator Y * of the process mean can be determined by the aid of the following formula (see, e.g. Cochran, 2007).
where ρ corresponds to the correlation coefficient between X and Y. The center line and the control limits of the two-sided monitoring scheme proposed by Haq (2020) where k is a positive design parameter which is determined appropriately in order to achieve a required degree of in-and out-of-control behavior. Haq (2020) illustrated the proposed chart for monitoring simulated bivariate data based on the Plackett's distribution.

Nonparametric EWMA Monitoring Schemes Based On Rank Statistics
In the present section, we refer to univariate nonparametric schemes, which rely on ranks and follow the general EWMA framework. It is evident that rank-based statistics have been already utilized for constructing Shewhart-type distribution-free control schemes (see, e.g. Triantafyllou and Panayiotou, 2020;Triantafyllou, 2021). However, our study aims at discussing the corresponding EWMA -type advances.
We next provide a detailed review of nonparametric univariate monitoring schemes based on ranks, namely of these members of the EWMANCC class which employ several rank-based statistics. The set-up of presentation of the included articles is determined to be chronological for the purpose of setting up a comprehensible and well-organized flux. Hackl and Ledolter (1991) introduced a nonparametric framework by considering the exponentially weighted moving means of the ranks of the observations. The resulting chart seems to be outlierresistant and performs well even if a sudden change of the process mean is occurred. Let , 1, 2,..., t Xt  be a sequence of independent random variables from a continuous distribution with cumulative distribution function F(x). The so-called standardized rank Rt of the observation Xt is defined as:

A Nonparametric EWMA Control Chart Based on Standardized Ranks
where 0 () Fx denotes the in-control distribution of the underlying production. As Hackl and Ledolter (1991) argued, the standardized rank Rt is uniformly distributed, while it takes on values between -1 and 1 with mean and variance equal to zero and 1/3 respectively. In case of an unknown in-control distribution, one may take advantage of an available historical in-control reference  Wilcoxon, 1945), one may arrange all (m+n) observations in ascending order and put them in a pooled sample. Afterwards, define ai such that: 1, if a observation is placed in the entry of the pooled sample 0, if a observation is placed in the entry of the pooled sample The Wilcoxon rank-sum statistic can now be viewed as: while its mean and variance under the identical distributions' assumption, are given by: (  (79) respectively. Based on the above-mentioned argumentation, Li et al. (2010) proposed the following EWMA-type statistic: where λ is the smoothing parameter of the EWMA scheme. The control limits of the resulting EWMA-type nonparametric monitoring scheme are given below: ( It goes without saying that the EWMA chart produces an out-of-control signal whenever i T falls outside or on the control limits. In addition, Malela-Majika et al. (2018) developed a nonparametric EWMA-type control chart based on the same rank statistic, namely the Wilcoxon rank-sum statistic. However, in their framework the so-called repetitive sampling is utilized in order to improve the sensitivity of the resulting EWMA scheme in tracking down plausible changes of the process mean.
The authors delivered theoretical formulae as well as practical algorithms to promote the structure and applicability of the resulting scheme. Moreover, its attribution is appraised by the aid of several characteristics, such as the ARL, the SDRL, the Average Sample Size or Percentiles of the Run Length.
Generally speaking, in a repetitive sampling plan, when a sample of magnitude n is drawn from the process, the number of sample points outside the control limits is supervised over a fixed criterion which corresponds to the upper (or lower) control limit of the resulting chart. The process is declared to be in-control, out-of-control or inconclusive when decision on the process cannot be made on the basis of the first sample (see, e.g. Sherman, 1965;Balamurali and Jun, 2006;Aslam et al., 2014).
The control chart proposed by Malela-Majika et al. (2018) utilizes the monitoring EWMA-type statistic defined in Eq. (18) and its construction calls for four different control limits. The one pair of limits, say LCL1, UCL1 correspond to the outer limits, while the others LCL2, UCL2 express the so-called inner limits of the resulting chart. The control limits of the proposed chart are given below: , the next step is to draw another sample of the same magnitude form the process and calculate the EWMA statistic once again. It is straightforward that if we set L1 = L2, then the monitoring scheme proposed by Malela-Majika et al. (2018) reduces to the one introduced by Li et al. (2010), whose presentation took place earlier.
Moreover, Malela-Majika (2021) considered a distribution-free EWMA-type monitoring framework, where the random variable W defined earlier is applied twice. The monitoring statistic of the proposed scheme can be viewed as: where, Note that Tt simply corresponds to the charting statistic of the scheme introduced by Li  ( 1) (2 2 ) ( 1) , 2 (2 ) 12 ( 1) (2 2 ) ( 1) 2 (2 ) 12 n m n mn m n LCL L n m n mn m n UCL L Finally, Letshedi et al. (2021) proposed a new distribution-free triple EWMA control chart based on the Wilcoxon rank-sum statistic to improve the ability for tracking down plausible changes in location parameter of the underlying distribution. In their framework, a new fast initial response feature is also activated in order to make the resulting chart more efficient. Following a similar argumentation as the one established by Alevizakos et al. (2020b), Letshedi et al. (2021) constructed their triple EWMA scheme based on the following monitoring statistic: where the statistic Zt is defined in Eq. (84).
The EWMA-type nonparametric control chart established by Letshedi et al. (2021) Liu et al. (2013) established sequential rank-based nonparametric adaptive EWMA monitoring scheme for tracking down the unceasing change in the location parameter. Suppose that the observations Xt are drawn independently over time from the following change-point model (see, e.g. Hawkins and Deng, 2010

A Sequential Rank-Based Distribution-Free Adaptive Ewma Monitoring Scheme
while τ is the unascertained change point parameter, μ0 (μ1) corresponds to the in-control (out-ofcontrol) location parameter of the process, while F denotes its continuous distribution function.
We next denote by Rt the t-th sequential rank, namely the rank of observed value xt among the observations x1, x2,…, xt. which can be viewed as: The mean and variance of Rt, under the presumption that the process is in-control, are given by: respectively. Since the distribution of t R varies as t increases, it seems that the corresponding standardized values should be considered. By defining the t-th standardized sequential rank as: Liu et al. (2013) introduced their adaptive nonparametric EWMA control chart, whose monitoring statistic is given by: where the initial value equal to zero. Note that weight η is actually an adaption factor and is determined via the following:   R  values and k, ω, λ are predetermined constants. For illustration purposes, the above-mentioned authors implemented their proposed nonparametric control chart for monitoring real data from an aluminum electrolytic capacitor manufacturing production. Liu et al. (2015) introduced a nonparametric EWMA control scheme utilizing the so-called variable sampling interval for tracking down a range of changes in the location parameter of the underlying process. In their framework, the sampling interval between each pair of samples is not fixed but rather depends on what is observed in the first sample. The general idea comes from the work provided by Reynolds et al. (1988), who considered that the time interval until the next group should be short if a sample reveals some indication of a change in the process and long if there is no suspicion of a change. Consequently, Liu et al. (2015) constructed their scheme by using s a short sampling interval if the charting statistic is close to but not actually outside the control limits and a long sampling interval if it is close to target. Liu et al. (2015) utilized the monitoring statistic defined in Eq. (92) and defined the sampling time interval Di as: where w is a warning limit, which is used for determining whether the sampling interval should be shortened or not, while h is the control limit, which formulates the so-called action region of the underlying process. The authors appraised the attribute of the resulting EWMA monitoring scheme by the aid of well-known measures, such as the expected value of time from the beginning to the point when the chart signals (even if the process is in-control) and the expected value of the time from process change to the time when the chart signals.

An Adaptive Distribution-Free EWMA Control Scheme with Variable Sampling Interval
In addition, Liu et al. (2021) proposed an adaptive nonparametric EWMA control chart with dynamic sampling intervals. The particular scheme can efficiently track down various magnitudes of changes and exhibits robust attribution under alternative distributional models (based on plenty simulation numerical results carried out by the authors). Moreover, the manageable computational workload confirms that this approach is applicable for monitoring big data flows. The EWMA-type monitoring scheme introduced by Liu et al. (2021) utilizes standardized sequential ranks, the test statistic defined in Eq. (92) and the so-called dynamic sampling intervals method, which does not call for specifying sampling intervals ahead of time, and may accomplish a roughly optimal attribution compared to other schemes in which the sampling intervals are presumed be fixed values. Mukherjee (2017) established nonparametric EWMA control charts for simultaneously supervising the location and scale parameters of the underlying process. The resulting chart relies on the utilization of the Lepage statistic, which is actually a mixture of the well-known Wilcoxon rank sum statistics T1 and the Ansari-Bradley test statistic T2 (see, e.g. Lepage, 1971 (1 ) , 1, 2,...

Nonparametric EWMA Control Schemes Based on Lepage-Type Statistics
The one-sided EWMA-type nonparametric control chart proposed by Mukherjee (2017) produces an out-control signal whenever the monitoring statistic * j Z is greater than the upper control limit UCL. Note that UCL could be either steady-state or time-varying depending on j.
In addition, Song et al. (2020b) Song et al. (2020a) introduced the concept of an EWMA monitoring scheme with a dynamic fast initial feature (FIR, hereafter) that restricts initial false alarms and also improves the out-of-control attribution of the resulting framework. More precisely, they established distributionfree FIR-based EWMA monitoring frameworks which rely on either the Lepage or the Cucconi statistic. The resulting charts track down changes little later than the conventional EWMA schemes when the process change occurs at a later stage and not in the early stage of monitoring. However, the proposed FIR-based nonparametric control charts seem to be beneficial for tracking down a process deviation at the zero-state or for a change at an early stage of monitoring. Huang et al. (2018) established two nonparametric EWMA control charts for joint monitoring of time between events and event magnitude. Their proposed schemes (JK chart and MA chart, International Journal of Mathematical, Engineering and Management Sciences Vol. 6, No. 3, 708-751, 2021 https://doi.org/10.33889/IJMEMS.2021.6.3.044 742 hereafter) are based on the so-called Mathur statistic and Jurečková and Kalina statistic respectively (Mathur, 2009;Jurečková and Kalina, 2012

Nonparametric EWMA Control Schemes for Monitoring Time Between Events and Event Magnitude
Consequently, Huang et al. (2018) utilized the following statistic: where Wi corresponds to the standardized value of Ui. The EWMA-type monitoring statistic of the proposed JK chart is given by: Both EWMA schemes established by Huang et al. (2018) perform better than their corresponding Shewhart schemes especially for slight to middling changes. The resulting JK scheme performs more efficiently than the corresponding MA chart in most cases apart from the case when location changes of asymmetric distributions come from same directions. Moreover, the MA EWMA chart meets some ARL -biased issues when the changes come from different directions. By the aid of an extensive simulation investigation, Huang et al. (2018) deduced that the JK chart is sensitive in monitoring different combinations of changes with same or different directions and still performs well even for highly tailed distributions and highly asymmetric distributions. Mukherjee et al. (2019) introduced EWMA-type distribution-free control schemes based on the socalled Hogg-Fisher-Randles (HFR, hereafter) statistic in order to investigate slight and tenacious changes in particular measurements of the underlying production. Gastwirth (1965) increased the efficiency of rank-based testing procedures by sacrificing some ranks of the combined samples. Based on this idea, Hogg et al. (1975) developed the so-called HFR statistic, which could be viewed as an alternative to the Wilcoxon rank-sum statistic defined earlier, for addressing the two-sample location problem, especially in the case of heavy-tailed distributions. Note that the abovementioned HFR test statistic has been already utilized in the literature (see, e.g. Mukherjee and Sen, 2015).

A Nonparametric EWMA Control Chart Based on Hogg-Fisher-Randle Statistic
Let 12 , ,..., m X X X be a sample of magnitude m drawn randomly from the process when it is incontrol. The X-observations form the so-called reference sample of the charting procedure. Afterwards, a lot of test samples of magnitude n 12 , ,..., , 1, 2,...
are drawn from the process in order to decide whether the process is still in-control or a change in either the location or the scale parameter has been occurred. Denoting by Rij the rank of observations Yij in the combined group of magnitude (m+n), the HFR statistic ΤHFR,i can be computed through the next formula: where [x] corresponds to the greatest integer contained in x. Since Hogg et al. (1975) considered the sum of ranks of only those Y samples, whose ranks are less than or equal to the median rank of the combined samples, the HFR statistic seems to be suitable for tracking down plausible changes in location parameter in right-skewed distributions.