# Unit Interval Time and Magnitude Monitoring Using Beta and Unit Gamma Distributions

Authors: Muhammad Farhan Akram; Sajid Ali; Ismail Shah; Giulia Marcon
Journal: Journal of Mathematics
 (2022)

---

## Abstract

Quick detection of an assignable cause is necessary for process accuracy with respect to the specifications. The aim of this study is to monitor the time and magnitude processes based on unit-interval data. To this end, maximum exponentially weighted moving average (Max-EWMA) control chart for simultaneous monitoring time and magnitude of an event is proposed. To be precise, beta and unit gamma distributions are considered to develop the Max-EWMA chart. The chart’s performance is accessed using average run length (ARL), the standard deviation of run length (SDRL), and different quantiles of the run length distribution through extensive Monte Carlo simulations. Besides a comprehensive simulation study, the proposed charting methodology is applied to a real data set. The results show that the proposed chart is more efficient in detecting small to medium-sized shifts. The results also indicate that simultaneous shifts are detected more quickly as compared to the pure shift.


---

## Body

## 1. Introduction

Statistical quality control (SQC) is the science of using statistical techniques in maintaining and monitoring the quality level of products and services. There are two main components of SQC: acceptance sampling and statistical process control (SPC). Acceptance sampling is a technique of selecting a lot (of products, goods, and raw material) based on the result (or quality) of selected samples. On the other hand, SPC uses sampling along with other statistical methods to assess the quality of output of a manufacturing process. A graphical representation known as the control chart of the process helps to determine whether the process output achieves the acceptable quality level or not.In SPC, monitoring of a process to check if it is in-control (IC) or out-of-control (OOC) is done only after the stable process parameters, including mean, fraction nonconforming, or variance, are obtained. Control charts have long been and extensively used in the areas of manufacturing to monitor a process’ stability, and recently, their applications have extended beyond the manufacturing processes. For instance, control charts have been used to monitor the stability of stock price or the quality of a service. Furthermore, they are used in disease surveillance systems to assess whether a disease has reached to the pandemic level or not.One of the techniques used in SPC to monitor a process is the Shewhart approach where the process is termed as IC or OOC based on the result of a current sample. This technique usually performs better when we have large size disturbances in a process. Since this technique completely ignores the past information (because of which this scheme is also called memoryless charting scheme), it cannot detect small shifts in the process. Contrary to Shewhart approach, the memory-type charting schemes (such as cumulative sum (CUSUM) and exponentially weighted moving average (EWMA)) consider not only the current but also the past information about the process. This feature enables these charts to detect small shifts in the process.Many real-life settings involve data of rates or proportions, e.g., monitoring the ratio of ingredients of life-saving drugs, etc., and hence, there arises a need to develop the charting mechanism for such types of data. The data of rates and proportions may not be resulted from a Bernoulli process, and therefore, control charts other than the traditional attribute charts, such as p and np charts, are required. For instance, monitoring the rate of unemployment to ensure the better utilization of resources in public politics can be performed with these charts. While dealing with different processes, magnitude associated with an event occurrence plays an important role to define its quality. This study aims at presenting the simultaneous monitoring methodology of time and magnitude based on the distributions having support between 0 and 1. To this end, the exponentially weighted moving average (EWMA) charting mechanism with a slight modification is used in order to develop the proposed chart. Different shift sizes are introduced, and the average speed, known as the average run length (ARL) to measure the average of the out-of-control signals, is used. The average run length is the most widely and frequently used criterion for evaluating the performance of a control chart. Our proposed chart is flexible and can be applied to many real-life scenarios.For Bernoulli data, attribute control charts such as p chart is applied to monitor the fraction and the number of nonconforming items, respectively [1]. An advantage of an attribute control chart over variable control chart is that it is economical and less time-consuming compared to the former because attribute data collection is much easier than variable data. Ho and Quinino [2] used an attribute control chart to monitor the variability of a process. There are a number of practical situations where one might be interested in the monitoring of proportions or rates in a product; however, the available data set may not follow the Bernoulli distribution [3]. The data of proportions and rates can be described as collection of observations where the variable of interest is confined to an interval of (0,1). Such data can be found extensively in many fields of study and often analyzed using normal distribution because of simplicity associated with normal distribution, yet misleading conclusions [4].Certain distributions are available in the literature which can be utilized to deal with proportion data. One of such distributions is known as the beta distribution. Sant’Anna and Ten Caten [5] considered this distribution along with Shewhart strategy to construct the beta control chart for monitoring fractional type variables of a manufacturing process. Later, Lima-Filho et al. [6] constructed an inflated beta control chart. Apart from beta distribution, the unit gamma (UG) can exhibit constant, unimodal, increasing, and decreasing shapes, whereas its hazard function has bathtub and decreasing shapes [7]. Grassia [8] gave a detailed overview of this distribution as well as of its variants. Ratnaparkhl and Mosimann [9] studied this distribution to derive some new distributions. The UG distribution is similar to the beta distribution in terms of some of the properties, and hence, some of its applications are the same as that of the beta distribution. This distribution can be used to estimate the density of virus or bacteria. Tadikamalla [10] showed that the UG distribution can be used as an alternative for Johnson SB and beta distributions. Recently, Mousa et al. [11] developed a regression model using this distribution.Time-between events (TBE) charts have recently been the focus of many researchers as these charts are very effective in monitoring high quality processes such as high yield production lines with extremely small defect rates [12]. Several TBE control charts are available in literature [13]. These charts also overcome the drawbacks of traditional Shewhart charting schemes, including high false alarm rate, the assumption of normality, and negative lower limit. Ali [14] proposed the TBE control chart based on the renewal process assuming a continuous exponentiated family of distributions. The authors concluded that the proposed chart is more efficient and flexible when compared to existing TBE charts based on exponential distribution. For some recent contributions in this direction, we refer to the works of Xie et al. [15], Zhang et al. [16], Shafae et al. [17], Shah et al. [18], Talib et al. [19], and references cited therein.Wu et al. [20] developed a rate chart for monitoring time and magnitude simultaneously by using the ratio of magnitude and time as the plotting statistic. Practically, during a certain time period, the effect of presence of an event, i.e., the damage or disturbance it can cause, is determined by its frequency or by its magnitude. Thus, there arises a need to monitor both of these characteristics (frequency and magnitude) simultaneously. Taking this into account, Wu et al. [21] adopted a Shewhart control charting technique for monitoring the frequency and magnitude of an event simultaneously. In earlier studies, simultaneous monitoring was done using two separate charting statistics. This technique was later found to be less effective when compared to the charts based on single statistic except for some exceptional situations. Most of the work on simultaneous monitoring is based on Shewhart mechanism [22, 23]. However, Shewhart charts are of memoryless nature, and the historical information of the process is ignored as the new points are plotted [24]. As a result, these charts cannot detect small shifts in process parameters satisfactorily. Consequently, memory-type control charts such as exponentially weighted moving average chart (EWMA) and cumulative sum (CUSUM) chart are required.The EWMA chart is more sensitive as compared to the Shewhart chart when it comes to reflecting critical past information. In addition, unlike Shewhart and CUSUM schemes, the EWMA chart is not sensitive to nonnormality of data for smaller values of the smoothing parameter [25]. Most recently, Sanusi et al. [26] introduced the maximum EWMA (Max-EWMA) control chart to simultaneously monitor the frequency and magnitude of events where the frequency was assumed to follow the exponential and magnitude followed a gamma distribution.The rest of the article is organized as follows. The probability density functions of the beta and unit gamma distributions with transformed parameters to have the same means are discussed in Section2. Max-EWMA chart using the beta and unit gamma distributions is defined in Section 3. The charting procedure is also defined in the same section. Section 4 presents the in and out of control performance of the Max-EWMA chart. Real data examples are discussed in Section 5, while concluding remarks are given in Section 6.


## 2. Beta and Unit Gamma Distribution

For the proposed Max-EWMA control chart, we consider the beta and unit gamma distribution in this section to provide in-depth analysis of the unit data monitoring problem.


### 2.1. Beta Distribution

A random variableT follows a beta distribution with shape parameters δ>0 and γ>0, if its probability density function (PDF) is given by(1)ft|δ,γ=tδ−11−tγ−1Bδ,γ,0<t<1,where Bδ,γ=ΓδΓγ/Γδ+γ is called the beta function and Γδ=∫0∞ωδ−1e−ωdt. The mean and variance of T are(2)ET=μ=δδ+γ,VarT=δγδ+γ2δ+γ+1,respectively. Suppose, we transform the parameters in equation (1) as(3)δ=μϕ,γ=1−μϕ.Now, using equation (3), the PDF of the beta distribution in equation (1) takes the form(4)ft|δ,γ=tμϕ−11−t1−μϕ−1Bμϕ,1−μϕ,0<t<1.Furthermore, the mean and the variance ofT with transformed parameters are ET=μ and VarT=μ1−μ/ϕ+1, respectively.


### 2.2. Unit Gamma Distribution

A random variable X follows the unit gamma distribution with shape parameterτ>0 and rate parameter θ>0, i.e., X∼uGτ,θ, if it has the following PDF:(5)fx|τ,θ=θτΓτxθ−1log1xτ−1,0<x<1,The mean and variance associated with equation (5) are(6)EX=μ=θθ+1τ,VarX=θθ+2τ−θθ+12τ,respectively. Let us transform θ in the above PDF as θ=μ1/τ/1−μ1/τ. Then, the PDF of X takes the form(7)fx|τ,θ=μ1/τ/1−μ1/ττΓτxμ1/τ/1−μ1/τ−1log1xτ−1,0<x<1,where τ>0 and 0<μ<1. Under this parameterization, we have EX=μ and VarX=μ1/2−μ1/ττ−μ.


## 2.1. Beta Distribution

A random variableT follows a beta distribution with shape parameters δ>0 and γ>0, if its probability density function (PDF) is given by(1)ft|δ,γ=tδ−11−tγ−1Bδ,γ,0<t<1,where Bδ,γ=ΓδΓγ/Γδ+γ is called the beta function and Γδ=∫0∞ωδ−1e−ωdt. The mean and variance of T are(2)ET=μ=δδ+γ,VarT=δγδ+γ2δ+γ+1,respectively. Suppose, we transform the parameters in equation (1) as(3)δ=μϕ,γ=1−μϕ.Now, using equation (3), the PDF of the beta distribution in equation (1) takes the form(4)ft|δ,γ=tμϕ−11−t1−μϕ−1Bμϕ,1−μϕ,0<t<1.Furthermore, the mean and the variance ofT with transformed parameters are ET=μ and VarT=μ1−μ/ϕ+1, respectively.


## 2.2. Unit Gamma Distribution

A random variable X follows the unit gamma distribution with shape parameterτ>0 and rate parameter θ>0, i.e., X∼uGτ,θ, if it has the following PDF:(5)fx|τ,θ=θτΓτxθ−1log1xτ−1,0<x<1,The mean and variance associated with equation (5) are(6)EX=μ=θθ+1τ,VarX=θθ+2τ−θθ+12τ,respectively. Let us transform θ in the above PDF as θ=μ1/τ/1−μ1/τ. Then, the PDF of X takes the form(7)fx|τ,θ=μ1/τ/1−μ1/ττΓτxμ1/τ/1−μ1/τ−1log1xτ−1,0<x<1,where τ>0 and 0<μ<1. Under this parameterization, we have EX=μ and VarX=μ1/2−μ1/ττ−μ.


## 3. Max-EWMA Control Chart Using Beta and Unit Gamma Distributions

To jointly monitor unit interval time and magnitude, the Max-EWMA charting procedure is adopted. The plotting statistic of the Max-EWMA control chart consists of the maximum value of two independent EWMA statistics, both in absolute terms. For joint monitoring of the TBE (T) and magnitude (X) of the event, the maximum of absolute values of TBE and magnitude is plotted. When an OOC signal is observed, the chart has the capability to identify whether the shift has occurred in T or X.In particular, this study considers that the TBE follows the beta distribution,T∼Bδ,γ, and magnitude of the event follows the unit gamma distribution, X∼uGτ,θ. It is also assumed that the time between the occurrences of two successive disturbances and magnitude is independent. Furthermore, the process is assumed to start at time t=0, and the IC parameters are assumed to be δ0,γ0 and τ0,θ0. Let the shifts in the parameters of T and X, respectively, are represented as follows, δ=Δδδ0, γ=Δγγ0, τ=Δττ0, and θ=Δθθ0, respectively, where Δδ,Δγ,Δτ, and Δθ represent the shifts in the shape parameters of the beta distribution and shape and rate parameter of the unit gamma distribution, respectively. To this end, we have further considered two cases.


### 3.1. Case 1(a): Shifts in Shape Parameters of Beta and Unit Gamma Distribution

In the first case, shifts in the first shape parameterδ of the beta distribution and shape parameter τ of unit gamma distribution are considered. The process is termed as IC when both Δδ and Δτ are equal to one. The following two independent statistics are defined in the construction of the Max-EWMA chart(8)Ui=Xi−μμ1/2−μ1/ττ−μ,Vi=Ti−μμ1−μ/ϕ+1.The EWMA statistics, based onUi and Vi, are defined as(9)Yi=1−λeYi−1+λeUi,0<λe≤1,i=1,2,…,Zi=1−λeZi−1+λeVi,0<λe≤1,i=1,2,…,where λe∈0,1 is one of the parameters of the EWMA control chart called the smoothing constant (the other parameter is known as the constant multiplier L). The starting values of two EWMA statistics are Y0=EUi=0 and Z0=EVi=0, respectively. Traditionally, a small value of λe is considered to be suitable for detecting small shifts, whereas a large value is deemed appropriate to detect large shifts. Practically, λe∈0.05,0.25 is considered to detect small or medium-sized shifts in the process [27].The monitoring statistic of the Max-EWMA control chart comprises the maximum of absolute ofYi and Zi, i.e.,(10)Mi=maxYi,Zi.If there is shift in either T (shape1 parameter of the beta distribution) or X (shape parameter of the unit gamma distribution) or in both parameters, theMi statistic becomes larger as compared to when there is no shift in either of the parameters. Furthermore, the proposed Max-EWMA chart only has the upper control limit (UCL) due to the fact that the plotting statistic is real and nonnegative number. The UCL is computed as(11)UCL=EM+LVarM.Here,E(M) and Var(M) are the IC values of mean and variance of the plotting statistic, and L is the constant multiplier that controls the width of control limit. When E(M) and Var(M) are unknown, one can replace them with sample mean and variance of M, calculated from the IC historical data. The Max-EWMA chart can detect the upward shift (UW) shift as well as the downward (DW) shift in the X and/or T simultaneously.


### 3.2. Case 1(b): Shifts in the Shape Parameter of Beta Distribution and Rate Parameter of the Unit Gamma Distribution

In the second case, shifts inγ parameter of the beta and θ parameter of the unit gamma distribution are considered. Similar to the first case, the process is IC when both Δγ and Δθ are equal to one.


### 3.3. Charting Procedure for the Max-EWMA Control Chart

It is of utmost importance to identify the variables (T and/or X) being the cause of the OOC signal once it is triggered by the chart. As mentioned previously, follow-up approach, once an OOC state has been established, can be labelled easily using the Max-EWMA chart. The summary of the construction of the Max-EWMA control chart for simultaneous monitoring of frequency and magnitude is given as follows:(1)
InitializeY0=Z0=0 and specify the desired ARL0 value. Also, select the suitable value of constant multiplier L and smoothing parameter λe to achieve the desired ARL0. Different values of L, for different combinations of τ,μ, and ϕ are given in Table 1. The table values are obtained assuming ARL0 370 and 500. The table also includes the standard deviation of average run length along with different percentiles to access the skewness of the run length distribution.(2)
Calculate the value of UCL using equation (11). If E(M) and Var(M) are not known, replace them with the corresponding sample values estimated from past data when the process does not contain any shift.(3)
Obtain theith test observation, i.e., obtain Xi,Ti, where Xi is ith observation of event’s magnitude and Ti is the corresponding TBE between the occurrences of event’s ith and i−1th observations.(4)
Forith test observation, compute the values of Mi,Ui,Vi,Yi, and Zi.(5)
Start monitoring the process by plotting theMi corresponding to the counter value i. The process is termed as the IC if Mi≤UCL, otherwise OOC. If the process is IC, monitoring continues; otherwise, proceed to the next step.(6)
Investigate the process to figure out if the process is really in an OOC state, if yes, take the necessary actions to eliminate the cause(s) behind an OOC signal. Resume the process of monitoring and move to step 3.(7)
To identify the OOC signal, label the points plotted above the UCL as per the symbols given in Table2. If both Yi and Zi exceed UCL, use the label “XT” to indicate shifts in both time and magnitude of the event. If only Yi exceeds UCL, use the label “X” to indicate a shift in the magnitude only. Similarly, if Zi alone exceeds UCL, use the label “T” to indicate a shift in TBE.Table 1
Constant multiplier and IC run length properties of max-EWMA control chart whenT∼Bδ,γ, X∼uGτ,θ, μ=0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50.
λeKARL0SDRL0q10q25q50q75q90KARL0SDRL0q10q25q50q75q90ARL0=370ARL0=500Case 1τ=155,ϕ=2900.052.718370.293357.63051.00114.00260.00507.00839.002.913500.901484.01866.90155.00352.00683.251127.000.103.050370.458362.65147.00114.00259.00503.25852.003.210499.010483.87260.00151.00350.00694.001139.000.153.195370.843362.01046.00115.00260.00505.00841.103.352500.784497.61659.00149.00349.00706.001140.100.203.222370.892368.25844.00110.00258.00512.00848.003.375497.327499.83055.00144.00344.00679.001143.200.253.329367.599368.56041.00105.00252.50504.00833.103.489497.583501.44955.00144.00345.00684.001150.000.303.384373.593368.58243.00113.75263.00512.00855.103.543500.492503.29055.00146.00343.00688.001138.100.353.394370.134370.49341.90108.75259.00514.00844.003.560502.520495.29058.00150.00351.00692.001151.100.403.407367.501365.59341.00106.00254.50510.00846.003.572500.229494.10555.00147.00346.00698.001142.000.453.461370.338362.49141.00109.00260.00514.00854.103.625500.339494.29055.00144.00349.00696.001150.000.503.456367.458370.28039.00104.00253.00509.00844.103.620497.380502.59652.00141.00342.00684.001155.00Case 2τ=96,ϕ=1480.052.713370.573352.68151.00116.00261.00511.00825.002.906500.045474.03369.00158.00357.00687.001142.000.103.026370.123360.81147.00112.00260.00508.00845.003.203500.490494.33061.00151.00350.00691.251142.200.153.182369.737360.04944.00112.00256.50512.00850.003.365500.976496.84856.00148.00347.00691.001145.000.203.260370.128368.28043.00112.00255.00512.00844.003.426497.290499.39355.00143.00341.00678.251139.000.253.330367.308365.29043.00109.00255.00500.00833.003.498497.280502.76851.00137.00340.00690.001159.000.303.375373.583371.48341.00107.00259.00520.00860.103.547500.285499.34354.00148.75348.00692.001152.000.353.429370.388369.19441.00109.00261.00508.00838.003.600498.295501.85054.00143.00338.00695.251156.100.403.462370.412371.49341.00109.00260.00507.00844.003.615500.128488.33557.00147.00350.00700.001144.000.453.501370.947365.89639.00107.00260.00514.00846.003.669503.895501.49253.00145.00349.00702.001164.000.503.505370.570365.21842.00108.00259.00516.00853.103.670499.749502.89451.00137.00347.00698.001147.10Case 3τ=51,ϕ=800.052.711370.786357.29751.00117.00262.00506.00831.102.909500.294484.82167.00155.00346.00683.001136.100.103.034370.837354.19749.00115.00267.00513.00842.003.210500.029484.46759.00149.00355.00695.251123.000.153.191369.015363.01645.00111.00255.00505.00850.003.368500.117491.44162.00150.00355.00687.001131.100.203.270372.558366.10443.00113.00260.00516.00846.003.447502.583501.44856.00144.00348.00690.001154.000.253.354371.859365.32742.00110.00261.00518.00848.003.524500.338484.78953.00144.00355.00694.001146.100.303.446368.284361.66841.00108.00257.00512.00844.003.632498.280502.53254.00142.00335.00696.001173.100.353.468370.782374.50340.00105.00258.00510.00850.003.648503.294497.30756.00149.00359.00696.001147.000.403.499367.339365.05840.00107.00353.50507.00854.003.687497.283497.10052.00144.00347.50691.001122.000.453.653368.480366.58342.00107.00256.00510.00830.003.822497.280493.68453.00143.00342.00689.001142.000.503.650373.943370.04841.00108.00263.00521.00850.103.838502.297495.78656.00147.00352.00707.001152.00Case 4τ=20,ϕ=310.052.722370.115357.00554.00119.00262.00506.00822.102.913499.731489.96770.00158.00346.50668.001131.100.103.048370.866359.02748.00115.00261.00513.00844.003.234501.045490.26160.00153.00350.00691.001143.000.153.236370.007363.04246.00113.75258.00506.00843.103.428501.583494.73657.00152.00351.00702.001139.100.203.369373.810365.39143.00111.00262.00520.00855.103.570502.204501.11557.00144.00347.00690.001153.100.253.467373.539365.29445.00112.00262.00521.00847.103.645500.476497.20655.00150.00349.00684.001144.000.303.646372.480367.10642.00110.00261.00521.00851.003.825498.294492.48053.90144.00349.00684.251149.000.353.654370.784369.28641.00108.00257.50514.25848.103.859500.240494.28557.00149.00353.50693.251147.000.403.731372.281370.03541.00108.00260.50517.00850.003.931502.574504.45852.00149.00350.00698.251153.100.453.761373.669374.10640.00107.00259.00517.00860.103.956500.119499.23055.00146.00346.00693.251147.100.503.800368.195364.06841.00108.00257.00510.00842.003.998503.251502.28955.00143.00350.00703.001164.10Table 2
Symbols representing the source of the out-of-control signal.
TimeMagnitudeY<UCLY>UCLZ<UCLICXZ>UCLTXT


## 3.1. Case 1(a): Shifts in Shape Parameters of Beta and Unit Gamma Distribution

In the first case, shifts in the first shape parameterδ of the beta distribution and shape parameter τ of unit gamma distribution are considered. The process is termed as IC when both Δδ and Δτ are equal to one. The following two independent statistics are defined in the construction of the Max-EWMA chart(8)Ui=Xi−μμ1/2−μ1/ττ−μ,Vi=Ti−μμ1−μ/ϕ+1.The EWMA statistics, based onUi and Vi, are defined as(9)Yi=1−λeYi−1+λeUi,0<λe≤1,i=1,2,…,Zi=1−λeZi−1+λeVi,0<λe≤1,i=1,2,…,where λe∈0,1 is one of the parameters of the EWMA control chart called the smoothing constant (the other parameter is known as the constant multiplier L). The starting values of two EWMA statistics are Y0=EUi=0 and Z0=EVi=0, respectively. Traditionally, a small value of λe is considered to be suitable for detecting small shifts, whereas a large value is deemed appropriate to detect large shifts. Practically, λe∈0.05,0.25 is considered to detect small or medium-sized shifts in the process [27].The monitoring statistic of the Max-EWMA control chart comprises the maximum of absolute ofYi and Zi, i.e.,(10)Mi=maxYi,Zi.If there is shift in either T (shape1 parameter of the beta distribution) or X (shape parameter of the unit gamma distribution) or in both parameters, theMi statistic becomes larger as compared to when there is no shift in either of the parameters. Furthermore, the proposed Max-EWMA chart only has the upper control limit (UCL) due to the fact that the plotting statistic is real and nonnegative number. The UCL is computed as(11)UCL=EM+LVarM.Here,E(M) and Var(M) are the IC values of mean and variance of the plotting statistic, and L is the constant multiplier that controls the width of control limit. When E(M) and Var(M) are unknown, one can replace them with sample mean and variance of M, calculated from the IC historical data. The Max-EWMA chart can detect the upward shift (UW) shift as well as the downward (DW) shift in the X and/or T simultaneously.


## 3.2. Case 1(b): Shifts in the Shape Parameter of Beta Distribution and Rate Parameter of the Unit Gamma Distribution

In the second case, shifts inγ parameter of the beta and θ parameter of the unit gamma distribution are considered. Similar to the first case, the process is IC when both Δγ and Δθ are equal to one.


## 3.3. Charting Procedure for the Max-EWMA Control Chart

It is of utmost importance to identify the variables (T and/or X) being the cause of the OOC signal once it is triggered by the chart. As mentioned previously, follow-up approach, once an OOC state has been established, can be labelled easily using the Max-EWMA chart. The summary of the construction of the Max-EWMA control chart for simultaneous monitoring of frequency and magnitude is given as follows:(1)
InitializeY0=Z0=0 and specify the desired ARL0 value. Also, select the suitable value of constant multiplier L and smoothing parameter λe to achieve the desired ARL0. Different values of L, for different combinations of τ,μ, and ϕ are given in Table 1. The table values are obtained assuming ARL0 370 and 500. The table also includes the standard deviation of average run length along with different percentiles to access the skewness of the run length distribution.(2)
Calculate the value of UCL using equation (11). If E(M) and Var(M) are not known, replace them with the corresponding sample values estimated from past data when the process does not contain any shift.(3)
Obtain theith test observation, i.e., obtain Xi,Ti, where Xi is ith observation of event’s magnitude and Ti is the corresponding TBE between the occurrences of event’s ith and i−1th observations.(4)
Forith test observation, compute the values of Mi,Ui,Vi,Yi, and Zi.(5)
Start monitoring the process by plotting theMi corresponding to the counter value i. The process is termed as the IC if Mi≤UCL, otherwise OOC. If the process is IC, monitoring continues; otherwise, proceed to the next step.(6)
Investigate the process to figure out if the process is really in an OOC state, if yes, take the necessary actions to eliminate the cause(s) behind an OOC signal. Resume the process of monitoring and move to step 3.(7)
To identify the OOC signal, label the points plotted above the UCL as per the symbols given in Table2. If both Yi and Zi exceed UCL, use the label “XT” to indicate shifts in both time and magnitude of the event. If only Yi exceeds UCL, use the label “X” to indicate a shift in the magnitude only. Similarly, if Zi alone exceeds UCL, use the label “T” to indicate a shift in TBE.Table 1
Constant multiplier and IC run length properties of max-EWMA control chart whenT∼Bδ,γ, X∼uGτ,θ, μ=0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50.
λeKARL0SDRL0q10q25q50q75q90KARL0SDRL0q10q25q50q75q90ARL0=370ARL0=500Case 1τ=155,ϕ=2900.052.718370.293357.63051.00114.00260.00507.00839.002.913500.901484.01866.90155.00352.00683.251127.000.103.050370.458362.65147.00114.00259.00503.25852.003.210499.010483.87260.00151.00350.00694.001139.000.153.195370.843362.01046.00115.00260.00505.00841.103.352500.784497.61659.00149.00349.00706.001140.100.203.222370.892368.25844.00110.00258.00512.00848.003.375497.327499.83055.00144.00344.00679.001143.200.253.329367.599368.56041.00105.00252.50504.00833.103.489497.583501.44955.00144.00345.00684.001150.000.303.384373.593368.58243.00113.75263.00512.00855.103.543500.492503.29055.00146.00343.00688.001138.100.353.394370.134370.49341.90108.75259.00514.00844.003.560502.520495.29058.00150.00351.00692.001151.100.403.407367.501365.59341.00106.00254.50510.00846.003.572500.229494.10555.00147.00346.00698.001142.000.453.461370.338362.49141.00109.00260.00514.00854.103.625500.339494.29055.00144.00349.00696.001150.000.503.456367.458370.28039.00104.00253.00509.00844.103.620497.380502.59652.00141.00342.00684.001155.00Case 2τ=96,ϕ=1480.052.713370.573352.68151.00116.00261.00511.00825.002.906500.045474.03369.00158.00357.00687.001142.000.103.026370.123360.81147.00112.00260.00508.00845.003.203500.490494.33061.00151.00350.00691.251142.200.153.182369.737360.04944.00112.00256.50512.00850.003.365500.976496.84856.00148.00347.00691.001145.000.203.260370.128368.28043.00112.00255.00512.00844.003.426497.290499.39355.00143.00341.00678.251139.000.253.330367.308365.29043.00109.00255.00500.00833.003.498497.280502.76851.00137.00340.00690.001159.000.303.375373.583371.48341.00107.00259.00520.00860.103.547500.285499.34354.00148.75348.00692.001152.000.353.429370.388369.19441.00109.00261.00508.00838.003.600498.295501.85054.00143.00338.00695.251156.100.403.462370.412371.49341.00109.00260.00507.00844.003.615500.128488.33557.00147.00350.00700.001144.000.453.501370.947365.89639.00107.00260.00514.00846.003.669503.895501.49253.00145.00349.00702.001164.000.503.505370.570365.21842.00108.00259.00516.00853.103.670499.749502.89451.00137.00347.00698.001147.10Case 3τ=51,ϕ=800.052.711370.786357.29751.00117.00262.00506.00831.102.909500.294484.82167.00155.00346.00683.001136.100.103.034370.837354.19749.00115.00267.00513.00842.003.210500.029484.46759.00149.00355.00695.251123.000.153.191369.015363.01645.00111.00255.00505.00850.003.368500.117491.44162.00150.00355.00687.001131.100.203.270372.558366.10443.00113.00260.00516.00846.003.447502.583501.44856.00144.00348.00690.001154.000.253.354371.859365.32742.00110.00261.00518.00848.003.524500.338484.78953.00144.00355.00694.001146.100.303.446368.284361.66841.00108.00257.00512.00844.003.632498.280502.53254.00142.00335.00696.001173.100.353.468370.782374.50340.00105.00258.00510.00850.003.648503.294497.30756.00149.00359.00696.001147.000.403.499367.339365.05840.00107.00353.50507.00854.003.687497.283497.10052.00144.00347.50691.001122.000.453.653368.480366.58342.00107.00256.00510.00830.003.822497.280493.68453.00143.00342.00689.001142.000.503.650373.943370.04841.00108.00263.00521.00850.103.838502.297495.78656.00147.00352.00707.001152.00Case 4τ=20,ϕ=310.052.722370.115357.00554.00119.00262.00506.00822.102.913499.731489.96770.00158.00346.50668.001131.100.103.048370.866359.02748.00115.00261.00513.00844.003.234501.045490.26160.00153.00350.00691.001143.000.153.236370.007363.04246.00113.75258.00506.00843.103.428501.583494.73657.00152.00351.00702.001139.100.203.369373.810365.39143.00111.00262.00520.00855.103.570502.204501.11557.00144.00347.00690.001153.100.253.467373.539365.29445.00112.00262.00521.00847.103.645500.476497.20655.00150.00349.00684.001144.000.303.646372.480367.10642.00110.00261.00521.00851.003.825498.294492.48053.90144.00349.00684.251149.000.353.654370.784369.28641.00108.00257.50514.25848.103.859500.240494.28557.00149.00353.50693.251147.000.403.731372.281370.03541.00108.00260.50517.00850.003.931502.574504.45852.00149.00350.00698.251153.100.453.761373.669374.10640.00107.00259.00517.00860.103.956500.119499.23055.00146.00346.00693.251147.100.503.800368.195364.06841.00108.00257.00510.00842.003.998503.251502.28955.00143.00350.00703.001164.10Table 2
Symbols representing the source of the out-of-control signal.
TimeMagnitudeY<UCLY>UCLZ<UCLICXZ>UCLTXT


## 4. Performance of the Proposed Chart

The average run length (ARL) is the most common measure to access the performance of a control chart. It is defined as the average of run length, i.e., average number of subgroups (samples) plotted until an OOC signal is observed. Two types of ARL are defined in the literature: the in-control ARLARL0 and the out-of-control ARL ARL1. Under ideal conditions, ARL0 should be as large as possible so that unnecessary pauses in the production line can be avoided. On the other hand, the smaller the value of ARL1, the better the performance of the control chart. In fact, in comparative analysis, the chart with a smaller value of ARL1 is considered the best. Since the ARL distribution is skewed, we need to study its standard deviation (SDRL) as well as the percentiles (q10, q25, q50, q75, and q90) of the run length. Therefore, a detailed analysis of the ARL along with its standard deviation (SDRL) and different percentiles of run length distribution are given in Tables 3 and 4 and Table S1 (Supplementary Materials) corresponding to Case 1(a), and in Tables S2–S4 corresponding to Case 1(b). Even though our interest lies in the simultaneous monitoring of the magnitude and TBE, shifts in a single parameter of each distribution, keeping IC the corresponding parameter of other distribution, is also considered. For example, shift in shape 1 parameter of the beta distribution is considered in Table 3 keeping the shape parameter of the unit gamma distribution in-control. Furthermore, shift in shape parameter of unit gamma, keeping IC the shape1 parameter of the beta distribution, is considered in Table 4. Finally, simultaneous shifts in shape1 parameter of the beta and shape parameter of unit gamma distribution is considered in Table S1. Similarly, for Case 1(b), a shift in shape 2 parameter of the beta distribution, keeping IC the rate parameter of the unit gamma distribution is considered in Table S2. Shift in the rate parameter of the unit gamma distribution, keeping IC the shape 2 parameter of the beta distribution, is considered in Table S3. Finally, simultaneous shifts in both the parameters are considered in Table S4.Table 3
Max-EWMA results considering shift in the shape1 parameter of beta distributionΔδ∈1.10,1.1,1.2,1.3,1.4,0.9,0.8,0.7,0.6,0.5,Δτ=1.0 when T∼Bδ,γ, X∼uGτ,θ, μ = 0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15.
λeKEMVMΔδΔτARLSDRLq10q25q50q75q90KARLSDRLq10q25q50q75q90ARL0=370ARL0=500Case 1τ=155,ϕ=2900.052.7180.180 026 60.009 312 01.001.00370.293357.63051.00114.00260.00507.00839.002.913500.901484.01866.90155.00352.00683.251127.000.052.7180.180 026 60.009 312 01.101.0020.4979.97510.0013.0018.0025.0034.002.91322.01210.68111.0014.0020.0027.0036.000.052.7180.180 026 60.009 312 01.201.008.8242.8026.007.008.0010.0013.002.9139.2752.8806.007.009.0011.0013.000.052.7180.180 026 60.009 312 01.301.005.7411.4594.005.006.007.008.002.9135.9961.5114.005.006.007.008.000.052.7180.180 026 60.009 312 01.401.004.3540.9553.004.004.005.006.002.9134.5260.9843.004.004.005.006.000.052.7180.180 026 60.009 312 00.901.0019.5249.01210.0013.0018.0024.0031.002.91320.6389.38711.0014.0019.0025.0033.000.052.7180.180 026 60.009 312 00.801.008.0242.2475.006.008.009.0011.002.9138.3652.3176.007.008.0010.0011.000.052.7180.180 026 60.009 312 00.701.005.0411.0604.004.005.006.006.002.9135.2571.1024.004.005.006.007.000.052.7180.180 026 60.009 312 00.601.003.7030.6633.003.004.004.004.002.9133.8510.6743.003.004.004.005.000.052.7180.180 026 60.009 312 00.501.002.9750.4212.003.003.003.003.002.9133.0760.4023.003.003.003.004.000.103.0400.258 086 10.019 059 31.001.00370.458362.65147.00114.00259.00503.25852.003.210499.010483.87260.00151.00350.00694.001139.000.103.0400.258 086 10.019 059 31.101.0020.87812.8948.0012.0018.0027.0038.003.21022.21013.7089.0012.0019.0028.0040.000.103.0400.258 086 10.019 059 31.201.007.7032.9764.006.007.009.0012.003.2107.9663.0945.006.007.0010.0012.000.103.0400.258 086 10.019 059 31.301.004.8051.4523.004.005.006.007.003.2105.0021.4993.004.005.006.007.000.103.0400.258 086 10.019 059 31.401.003.5880.9343.003.003.004.005.003.2103.7070.9513.003.004.004.005.000.103.0400.258 086 10.019 059 30.901.0019.66811.4818.0012.0017.0025.0035.003.21021.26012.7509.0012.0018.0026.0038.000.103.0400.258 086 10.019 059 30.801.006.8832.3294.005.006.008.0010.003.2107.1922.4435.005.007.008.0010.000.103.0400.258 086 10.019 059 30.701.004.1751.0543.003.004.005.006.003.2104.3061.0803.004.004.005.006.000.103.0400.258 086 10.019 059 30.601.003.0250.6262.003.003.003.004.003.2103.1270.6302.003.003.003.004.000.103.0400.258 086 10.019 059 30.501.002.3480.4862.002.002.003.003.003.2102.4310.5102.002.002.003.003.000.153.1950.320 219 40.029 386 71.001.00370.843362.01046.00115.00260.00505.00841.103.352500.784497.61659.00149.00349.00706.001140.100.153.1950.320 219 40.029 386 71.101.0022.76116.3397.0011.0018.0030.0044.003.35225.13218.3988.0012.0020.0033.0049.000.153.1950.320 219 40.029 386 71.201.007.3263.3224.005.007.009.0012.003.3527.7513.5104.005.007.009.0012.000.153.1950.320 219 40.029 386 71.301.004.3771.4953.003.004.005.006.003.3524.5811.5423.003.004.005.007.000.153.1950.320 219 40.029 386 71.401.003.2130.9172.003.003.004.004.003.3523.3070.9322.003.003.004.004.000.153.1950.320 219 40.029 386 70.901.0021.76614.9878.0011.0018.0028.0041.003.35223.88016.7228.0012.0019.0031.0046.000.153.1950.320 219 40.029 386 70.801.006.4902.5824.005.006.008.0010.003.3526.8382.7064.005.006.008.0010.000.153.1950.320 219 40.029 386 70.701.003.7651.0583.003.004.004.005.003.3523.8841.0803.003.004.004.005.000.153.1950.320 219 40.029 386 70.601.002.6790.6382.002.003.003.003.003.3522.7730.6472.002.003.003.004.000.153.1950.320 219 40.029 386 70.501.002.1290.3482.002.002.002.003.003.3522.1730.3862.002.002.002.003.00Case 2τ=96,ϕ=1480.052.7130.180 650 10.009 289 11.001.00370.559360.57851.00116.00259.50511.00835.102.906500.162485.59971.00156.00353.00683.001112.000.052.7130.180 650 10.009 289 11.101.0033.48420.28414.0019.0028.0042.0060.002.90636.01221.60615.0021.0031.0045.0064.000.052.7130.180 650 10.009 289 11.201.0013.1555.0778.009.0012.0016.0020.002.90614.0095.5118.0010.0013.0017.0021.000.052.7130.180 650 10.009 289 11.301.008.2802.5865.006.008.0010.0012.002.9068.7162.6796.007.008.0010.0012.000.052.7130.180 650 10.009 289 11.401.006.1411.6244.005.006.007.008.002.9066.4341.6684.005.006.007.009.000.052.7130.180 650 10.009 289 10.901.0031.50218.05114.0019.0027.0039.0055.002.90634.11619.49315.0020.0029.0043.0060.000.052.7130.180 650 10.009 289 10.801.0011.8204.1167.009.0011.0014.0017.002.90612.5374.3188.009.0012.0015.0018.000.052.7130.180 650 10.009 289 10.701.007.1671.8395.006.007.008.0010.002.9067.4971.8885.006.007.009.0010.000.052.7130.180 650 10.009 289 10.601.005.1531.0634.004.005.006.007.002.9065.3441.0754.005.005.006.007.000.052.7130.180 650 10.009 289 10.501.004.0180.7073.004.004.004.005.002.9064.1910.7193.004.004.005.005.000.103.0260.259 390 10.019 083 31.001.00370.123360.81147.00112.00260.00508.00845.003.207500.490494.33061.00151.00350.00691.251142.200.103.0260.259 390 10.019 083 31.101.0036.56727.53311.0017.0029.0048.0072.003.20741.29731.51912.0019.0032.5054.0081.000.103.0260.259 390 10.019 083 31.201.0012.2356.1206.008.0011.0015.0020.003.20712.8546.4566.008.0011.0016.0021.000.103.0260.259 390 10.019 083 31.301.007.1812.7294.005.007.009.0011.003.2077.5402.8914.005.007.009.0011.000.103.0260.259 390 10.019 083 31.401.005.1891.6503.004.005.006.007.003.2075.3901.7103.004.005.006.008.000.103.0260.259 390 10.019 083 30.901.0035.67325.67112.0018.0028.0046.0069.003.20740.48730.12413.0019.0032.0052.0080.000.103.0260.259 390 10.019 083 30.801.0010.7894.7246.007.0010.0013.0017.003.20711.4055.0256.008.0010.0014.0018.000.103.0260.259 390 10.019 083 30.701.006.0771.8774.005.006.007.009.003.2076.3851.9844.005.006.007.009.000.103.0260.259 390 10.019 083 30.601.004.2371.0333.004.004.005.006.003.2074.4241.0803.004.004.005.006.000.103.0260.259 390 10.019 083 30.501.003.2910.6543.003.003.004.004.003.2073.4060.6773.003.003.004.004.000.153.1820.320 924 50.029 420 41.001.00369.737360.04944.00112.00256.50512.00850.003.365500.976496.84856.00148.00347.00691.001145.000.153.1820.320 924 50.029 420 41.101.0042.33835.52310.0017.0032.0056.0087.003.36548.99741.60411.0020.0037.0065.00103.000.153.1820.320 924 50.029 420 41.201.0012.2607.3065.007.0010.0015.0022.003.36513.2397.8716.008.0011.0017.0023.000.153.1820.320 924 50.029 420 41.301.006.8042.9954.005.006.008.0011.003.3657.1513.1374.005.007.009.0011.000.153.1820.320 924 50.029 420 41.401.004.7611.7073.004.004.006.007.003.3654.9521.7753.004.005.006.007.000.153.1820.320 924 50.029 420 40.901.0044.03936.56411.0018.0033.0058.0091.003.36551.92743.70713.0021.0039.0069.00109.000.153.1820.320 924 50.029 420 40.801.0010.7785.4945.007.0010.0013.0018.003.36511.5606.0466.007.0010.0014.0019.000.153.1820.320 924 50.029 420 40.701.005.6752.0334.004.005.007.008.003.3655.9432.0934.004.006.007.009.000.153.1820.320 924 50.029 420 40.601.003.8671.0663.003.004.004.005.003.3654.0171.1143.003.004.005.005.000.153.1820.320 924 50.029 420 40.501.002.9350.6962.002.003.003.004.003.3653.0390.6962.003.003.003.004.00Case 3τ=51,ϕ=800.052.7110.180 491 70.009 324 51.001.00370.786357.29751.00117.00262.00506.00831.102.909500.195484.82167.00155.00346.00683.001136.100.052.7110.180 491 70.009 324 51.101.0053.00537.11118.0027.0043.0069.00102.102.90857.89541.61619.0028.0046.0075.00112.000.052.7110.180 491 70.009 324 51.201.0019.6829.47410.0013.0018.0024.0032.002.90821.01410.09710.0014.0019.0026.0034.000.052.7110.180 491 70.009 324 51.301.0011.9744.5657.009.0011.0014.0018.002.90812.5134.7237.009.0012.0015.0019.000.052.7110.180 491 70.009 324 51.401.008.6592.7926.007.008.0010.0012.002.9089.0652.8866.007.009.0011.0013.000.052.7110.180 491 70.009 324 50.901.0050.71634.34018.0027.0041.0065.0096.002.90857.80039.62820.0030.0047.0075.00110.000.052.7110.180 491 70.009 324 50.801.0017.5517.44210.0012.0016.0021.0027.002.90818.8138.13510.0013.0017.0023.0029.000.052.7110.180 491 70.009 324 50.701.0010.2923.1797.008.0010.0012.0014.002.90810.7083.3077.008.0010.0012.0015.000.052.7110.180 491 70.009 324 50.601.007.1071.7395.006.007.008.009.002.9087.4871.8215.006.007.008.0010.000.052.7110.180 491 70.009 324 50.501.005.4551.1044.005.005.006.007.002.9085.7051.1334.005.006.006.007.000.103.0340.259 294 40.019 150 01.001.00370.837354.19749.00115.00267.00513.00842.003.210500.029484.46759.00149.00355.00695.251123.000.103.0340.259 294 40.019 150 01.101.0061.76751.60615.0025.0047.0081.25130.103.21072.34561.31817.0029.0054.0097.00152.000.103.0340.259 294 40.019 150 01.201.0019.62211.8768.0011.0017.0025.0035.003.21020.93112.9558.0012.0018.0026.0038.000.103.0340.259 294 40.019 150 01.301.0010.8085.0725.007.0010.0013.0017.003.21011.4405.3606.008.0010.0014.0019.000.103.0340.259 294 40.019 150 01.401.007.5342.9744.005.007.009.0011.003.2107.8573.0255.006.007.009.0012.000.103.0340.259 294 40.019 150 00.901.0066.02753.78617.0028.0050.0088.00136.003.21078.14966.74019.0031.0058.00104.00165.000.103.0340.259 294 40.019 150 00.801.0017.4079.5498.0011.0015.0022.0030.003.21018.68810.3668.0011.0016.0023.0032.00>0.103.0340.259 294 40.019 150 00.701.009.1373.5035.007.008.0011.0014.003.2109.5813.6946.007.009.0011.0015.000.103.0340.259 294 40.019 150 00.601.006.0881.8234.005.006.007.008.003.2106.3171.8584.005.006.007.009.000.103.0340.259 294 40.019 150 00.501.004.5451.1043.004.004.005.006.003.2104.7361.1244.004.005.005.006.000.153.1910.321 317 50.029 541 01.001.00369.015363.01645.00111.00255.00505.00850.003.368500.117491.44162.00150.00355.00687.001131.100.153.1910.321 317 50.029 541 01.101.0071.64264.78614.0026.0052.0095.00157.003.36885.58777.54316.0030.0062.00118.00188.000.153.1910.321 317 50.029 541 01.201.0020.94814.8447.0011.0017.0027.0040.003.36823.22016.9997.0011.0019.0030.0045.000.153.1910.321 317 50.029 541 01.301.0010.7655.9095.007.009.0013.0018.003.36811.4936.3845.007.0010.0014.0020.000.153.1910.321 317 50.029 541 01.401.007.1823.2744.005.007.009.0011.003.3687.5163.4254.005.007.009.0012.000.153.1910.321 317 50.029 541 00.901.0088.24580.19817.0032.0064.00119.00193.003.368110.05798.92620.0038.0080.00151.00241.000.153.1910.321 317 50.029 541 00.801.0019.15712.6987.0010.0016.0024.0036.003.36821.08414.0028.0011.0017.0027.0039.000.153.1910.321 317 50.029 541 00.701.008.8473.9915.006.008.0011.0014.003.3689.4194.1835.006.009.0011.0015.000.153.1910.321 317 50.029 541 00.601.005.6501.9214.004.005.007.008.003.3685.9191.9934.004.006.007.008.000.153.1910.321 317 50.029 541 00.501.004.1211.0963.003.004.005.006.003.3684.2991.1683.003.004.005.006.00Case 4τ=20,ϕ=310.052.7220.180 379 60.009 286 71.001.00371.624355.93652.00115.00264.00510.00839.002.913499.713475.50569.00157.00359.00678.001123.000.052.7220.180 379 60.009 286 71.101.00102.89087.21925.0042.0077.00134.00216.002.913118.797101.98026.9047.0088.00158.00254.000.052.7220.180 379 60.009 286 71.201.0038.21523.90015.0021.0032.0049.0069.102.91342.07227.10416.0023.0035.0054.0077.000.052.7220.180 379 60.009 286 71.301.0021.93911.43710.0014.0019.0027.0037.002.91323.34711.97311.0015.0021.0029.0039.000.052.7220.180 379 60.009 286 71.401.0015.2026.5318.0011.0014.0019.0024.002.91316.2647.1279.0011.0015.0020.0026.000.052.7220.180 379 60.009 286 70.901.00109.60888.08528.0046.0083.00147.00224.002.913128.305105.21032.0054.0097.00170.00266.100.052.7220.180 379 60.009 286 70.801.0034.58919.80615.0021.0030.0043.0061.002.91337.45021.39116.0022.0032.0047.0065.000.052.7220.180 379 60.009 286 70.701.0018.4867.69610.0013.0017.0022.0029.002.91319.5498.11111.0014.0018.0024.0030.000.052.7220.180 379 60.009 286 70.601.0012.3234.0918.009.0012.0014.0018.002.91312.9094.2548.0010.0012.0015.0019.000.052.7220.180 379 60.009 286 70.501.009.0692.3996.007.009.0010.0012.002.9139.5912.4857.008.009.0011.0013.000.103.0480.259 130 80.019 243 31.001.00369.605360.08546.00113.00258.00510.25844.003.234501.045490.02659.00149.00349.00691.001144.100.103.0480.259 130 80.019 243 31.101.00121.821111.48522.0043.0088.00165.00267.003.234145.184135.10124.0049.00104.00195.00324.000.103.0480.259 130 80.019 243 31.201.0042.27633.25612.0019.0033.0055.0086.003.23449.54839.63613.0022.0038.0065.00101.000.103.0480.259 130 80.019 243 31.301.0022.33714.4788.0012.0019.0028.0041.003.23424.73416.2729.0013.0020.0032.0046.000.103.0480.259 130 80.019 243 31.401.0014.7168.0826.909.0013.0019.0025.003.23415.5988.5867.009.0014.0020.0027.000.103.0480.259 130 80.019 243 30.901.00166.689156.86628.0056.00118.00225.00373.003.234220.609204.47536.0074.00160.00302.00483.000.103.0480.259 130 80.019 243 30.801.0042.61331.08414.0020.0033.0056.0082.003.23449.08036.58215.0023.0039.0063.0097.000.103.0480.259 130 80.019 243 30.701.0018.70910.1809.0011.0016.0023.0032.003.23420.35311.2159.0012.0018.0025.0035.000.103.0480.259 130 80.019 243 30.601.0011.4854.7457.008.0010.0014.0018.003.23412.1314.9417.009.0011.0015.0019.000.103.0480.259 130 80.019 243 30.501.008.0592.6765.006.008.009.0012.003.2348.4282.7556.006.008.0010.0012.000.153.2360.321 317 50.029 5411.001.00370.007363.04246.00113.75258.00506.00843.103.428501.583494.73657.00152.00351.00702.001139.100.153.2360.321 317 50.029 5411.101.00132.713127.20020.0042.0094.00181.00300.003.428167.453159.60324.0054.00120.00230.00372.000.153.2360.321 317 50.029 5411.201.0049.44342.18311.0020.0037.0066.00105.003.42857.58649.86412.0022.0043.0077.00123.000.153.2360.321 317 50.029 5411.301.0024.50318.4028.0012.0019.0032.0048.003.42827.29921.3308.0013.0021.0035.0054.000.153.2360.321 317 50.029 5411.401.0015.21510.0336.008.0013.0019.0028.003.42816.46110.7096.009.0014.0021.0030.000.153.2360.321 317 50.029 5410.901.00248.646240.20833.0077.00174.00345.25563.003.428345.015338.70344.00104.00244.00475.00774.100.153.2360.321 317 50.029 5410.801.0058.91449.93414.0024.0044.0078.00123.003.42874.06664.87217.0029.0054.0098.00158.000.153.2360.321 317 50.029 5410.701.0021.71614.3598.0012.0018.0028.0041.003.42824.74616.6619.0013.0020.0032.0046.000.153.2360.321 317 50.029 5410.601.0011.8335.8706.008.0010.0014.0020.003.42812.6756.3726.008.0011.0015.0021.000.153.2360.321 317 50.029 5410.501.007.8242.9985.006.007.009.0012.003.4288.3033.2265.006.008.0010.0013.00Table 4
Max-EWMA results considering shift in the shape parameter of unit gamma distributionΔτ∈1.0,1.1,1.2,1.3,1.4,0.9,0.8,0.7,0.6,0.5,Δδ=1.0 when T∼Bδ,γ, X∼uGτ,θ, μ=0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15.
λeKEMVMΔδΔτARLSDRLq10q25q50q75q90KARLSDRLq10q25q50q75q90ARL0=370ARL0=500Case 1τ=155,ϕ=2900.052.7180.180 026 60.009 312 01.001.00370.028354.6651.00115.00262.00515.00827.002.913500.31485.3669.00158.00352.00689.001133.000.052.7180.180 026 60.009 312 01.001.1010.223 43.083 27.008.0010.0012.0014.002.91310.7083.231 97.008.0010.0012.0015.000.052.7180.180 026 60.009 312 01.001.205.119 30.978 84.004.005.006.006.002.9135.349 80.996 74.005.005.006.007.000.052.7180.180 026 60.009 312 01.001.303.675 30.573 73.003.004.004.004.002.9133.836 20.573 23.003.004.004.004.000.052.7180.180 026 60.009 312 01.001.403.046 40.282 93.003.003.003.003.002.9133.108 40.327 23.003.003.003.004.000.052.7180.180 026 60.009 312 01.000.908.477 42.834 65.006.008.0010.0012.002.9138.938 12.956 86.007.008.0011.0013.000.052.7180.180 026 60.009 312 01.000.803.784 70.888 73.003.004.004.005.002.9133.9360.905 83.003.004.004.005.000.052.7180.180 026 60.009 312 01.000.702.4060.523 32.002.002.003.003.002.9132.498 90.541 12.002.002.003.003.000.052.7180.180 026 60.009 312 01.000.601.899 10.326 41.002.002.002.002.002.9131.945 50.284 52.002.002.002.002.000.052.7180.180 026 60.009 312 01.000.501.3340.471 71.001.001.002.002.002.9131.423 50.494 11.001.001.002.002.000.103.040.258 086 10.019 059 31.001.00370.286 8360.7645.00109.00259.00516.00852.003.210499.25484.1663.00153.00354.00682.001138.000.103.040.258 086 10.019 059 31.001.109.092 13.394 55.007.008.0011.0014.003.2109.491 43.5426.007.009.0011.0014.000.103.040.258 086 10.019 059 31.001.204.236 80.958 33.004.004.005.005.003.2104.411 40.989 73.004.004.005.006.000.103.040.258 086 10.019 059 31.001.303.0210.499 22.003.003.003.004.003.2103.115 10.501 53.003.003.003.004.000.103.040.258 086 10.019 059 31.001.402.386 30.489 62.002.002.003.003.003.2102.503 70.505 82.002.003.003.003.000.103.040.258 086 10.019 059 31.000.907.408 32.994.005.007.009.0011.003.2107.744 63.1584.005.007.009.0012.000.103.040.258 086 10.019 059 31.000.803.085 70.837 42.003.003.004.004.003.2103.191 90.858 82.003.003.004.004.000.103.040.258 086 10.019 059 31.000.702.017 70.414 32.002.002.002.002.003.2102.060 90.411 62.002.002.002.003.000.103.040.258 086 10.019 059 31.000.601.4380.496 41.001.001.002.002.003.2101.507 90.500 81.001.002.002.002.000.103.040.258 086 10.019 059 31.000.501.037 20.189 31.001.001.001.001.003.2101.048 80.215 51.001.001.001.001.000.153.1950.320 924 50.029 386 71.001.00370.842 9362.0146.00115.00260.00505.00841.103.352500.784495.6858.00152.00351.50689.251133.100.153.1950.320 924 50.029 386 71.001.108.952 24.052 85.006.008.0011.0014.003.3539.361 14.185 65.006.008.0011.0015.000.153.1950.320 924 50.029 386 71.001.203.819 40.954 73.003.004.004.005.003.3533.956 10.983 13.003.004.004.005.000.153.1950.320 924 50.029 386 71.001.302.6660.562 72.002.003.003.003.003.3532.763 10.556 42.002.003.003.003.000.153.1950.320 924 50.029 386 71.001.402.112 90.316 82.002.002.002.003.003.3532.159 90.367 12.002.002.002.003.000.153.1950.320 924 50.029 386 71.000.907.045 33.283 64.005.006.009.0011.003.3537.359 83.486 54.005.007.009.0012.000.153.1950.320 924 50.029 386 71.000.802.769 40.823 82.002.003.003.004.003.3532.866 60.852 72.002.003.003.004.000.153.1950.320 924 50.029 386 71.000.701.799 90.485 31.002.002.002.002.003.3531.853 10.467 31.002.002.002.002.000.153.1950.320 924 50.029 386 71.000.601.203 70.402 81.001.001.001.002.003.3531.242 70.428 71.001.001.001.002.000.153.1950.320 924 50.029 386 71.000.501.005 90.076 61.001.001.001.001.003.3531.008 80.093 41.001.001.001.001.00Case 2τ=96,ϕ=1480.052.7130.180 650 10.009 289 11.001.00370.559358.57852.00115.00259.50511.00835.102.906500.162485.59971.00156.00353.00683.001112.000.052.7130.180 650 10.009 289 11.001.1013.6824.8618.0010.0013.0016.0020.002.90614.4625.0299.0011.0014.0017.0021.000.052.7130.180 650 10.009 289 11.001.206.5911.4285.006.006.007.008.002.9066.8691.4695.006.007.008.009.000.052.7130.180 650 10.009 289 11.001.304.6490.7624.004.005.005.006.002.9064.8290.7734.004.005.005.006.000.052.7130.180 650 10.009 289 11.001.403.7610.5523.003.004.004.004.002.9063.9180.5183.004.004.004.004.000.052.7130.180 650 10.009 289 11.000.9011.3824.4956.008.0011.0014.0017.002.90611.8894.5917.009.0011.0014.0018.000.052.7130.180 650 10.009 289 11.000.804.8031.2793.004.005.006.006.002.9064.9981.3263.004.005.006.007.000.052.7130.180 650 10.009 289 11.000.702.9960.6782.003.003.003.004.002.9063.1110.6942.003.003.003.004.000.052.7130.180 650 10.009 289 11.000.602.1580.4102.002.002.002.003.002.9062.2150.4352.002.002.002.003.000.052.7130.180 650 10.009 289 11.000.501.8010.4071.002.002.002.002.002.9061.8670.3581.002.002.002.002.000.103.0260.259 390 10.019 083 31.001.00370.864361.68648.00111.00260.00519.00847.003.207499.903488.66061.00154.00354.00678.251133.000.103.0260.259 390 10.019 083 31.001.1012.8415.8847.009.0012.0016.0020.003.20713.6206.2367.009.0012.0016.0022.000.103.0260.259 390 10.019 083 31.001.205.5371.4354.005.005.006.007.003.2075.7741.4824.005.006.007.008.000.103.0260.259 390 10.019 083 31.001.303.8030.7373.003.004.004.005.003.2073.9670.7563.003.004.004.005.000.103.0260.259 390 10.019 083 31.001.403.0810.4413.003.003.003.004.003.2073.1750.4503.003.003.003.004.000.103.0260.259 390 10.019 083 31.000.9010.1494.8845.007.009.0013.0017.003.20710.5515.0775.007.009.0013.0017.000.103.0260.259 390 10.019 083 31.000.803.9351.2353.003.004.005.006.003.2074.1121.2533.003.004.005.006.000.103.0260.259 390 10.019 083 31.000.702.4490.6182.002.002.003.003.003.2072.5240.6332.002.002.003.003.000.103.0260.259 390 10.019 083 31.000.601.8330.4381.002.002.002.002.003.2071.8890.4271.002.002.002.002.000.103.0260.259 390 10.019 083 31.000.501.3120.4631.001.001.002.002.003.2071.3770.4861.001.001.002.002.000.153.1820.321 317 50.029 420 41.001.00369.685365.81343.00111.00256.00511.00845.003.365500.280484.31057.00150.00351.00698.001136.000.153.1820.321 317 50.029 420 41.001.1013.4377.3826.008.0012.0017.0023.003.36214.6648.3557.009.0012.0018.0025.000.153.1820.321 317 50.029 420 41.001.205.1161.5243.004.005.006.007.003.3625.3721.6014.004.005.006.007.000.153.1820.321 317 50.029 420 41.001.303.4170.7233.003.003.004.004.003.3623.5710.7463.003.003.004.005.000.153.1820.321 317 50.029 420 41.001.402.7180.5392.002.003.003.003.003.3622.8420.5192.003.003.003.003.000.153.1820.321 317 50.029 420 41.000.909.8535.3944.006.009.0012.0017.003.36210.5325.9065.006.009.0013.0018.000.153.1820.321 317 50.029 420 41.000.803.5761.2412.003.003.004.005.003.3623.7161.2802.003.004.004.005.000.153.1820.321 317 50.029 420 41.000.702.1940.5942.002.002.003.003.003.3622.2560.6032.002.002.003.003.000.153.1820.321 317 50.029 420 41.000.601.5890.5101.001.002.002.002.003.3621.6480.5021.001.002.002.002.000.153.1820.321 317 50.029 420 41.000.501.1380.3451.001.001.001.002.003.3621.1670.3731.001.001.001.002.00Case 3τ=51,ϕ=800.052.7110.180 491 70.009 324 51.001.00370.377350.69752.00115.75262.00519.00835.002.909500.460483.26267.00155.00353.00688.001125.000.052.7110.180 491 70.009 324 51.001.1020.9239.19611.0014.0019.0025.0033.002.90922.4819.84812.0016.0020.0027.0035.000.052.7110.180 491 70.009 324 51.001.209.4112.4577.008.009.0011.0013.002.9099.8732.5867.008.009.0011.0013.000.052.7110.180 491 70.009 324 51.001.306.4651.2375.006.006.007.008.002.9096.7571.2625.006.007.007.008.000.052.7110.180 491 70.009 324 51.001.405.1370.7914.005.005.006.006.002.9095.3620.8144.005.005.006.006.000.052.7110.180 491 70.009 324 51.000.9016.8328.0048.0011.0015.0021.0027.002.90917.9528.4929.0012.0016.0022.0029.000.052.7110.180 491 70.009 324 51.000.806.7222.1644.005.006.008.0010.002.9096.9962.2084.005.007.008.0010.000.052.7110.180 491 70.009 324 51.000.704.0471.0603.003.004.005.005.002.9094.1951.0783.003.004.005.006.000.052.7110.180 491 70.009 324 51.000.602.8480.6812.002.003.003.004.002.9092.9530.6982.003.003.003.004.000.052.7110.180 491 70.009 324 51.000.502.1800.4312.002.002.002.003.002.9092.2470.4672.002.002.003.003.000.103.0340.259 294 40.019 150 01.001.00370.080361.08047.00115.00258.00509.00833.003.210500.197486.35061.00152.00352.50686.001124.000.103.0340.259 294 40.019 150 01.001.1022.25212.83710.0013.0019.0028.0039.003.21124.23914.51910.0014.0020.0030.0043.000.103.0340.259 294 40.019 150 01.001.208.2932.7055.006.008.0010.0012.003.2118.7952.8906.007.008.0010.0013.000.103.0340.259 294 40.019 150 01.001.305.4401.2564.005.005.006.007.003.2115.6691.3084.005.005.006.007.000.103.0340.259 294 40.019 150 01.001.404.2450.7743.004.004.005.005.003.2114.4010.8034.004.004.005.005.000.103.0340.259 294 40.019 150 01.000.9016.0179.4407.009.0014.0020.0028.003.21117.15310.1427.0010.0015.0021.0030.000.103.0340.259 294 40.019 150 01.000.805.6602.1393.004.005.007.009.003.2115.9672.2823.004.006.007.009.000.103.0340.259 294 40.019 150 01.000.703.3301.0122.003.003.004.005.003.2113.4471.0422.003.003.004.005.000.103.0340.259 294 40.019 150 01.000.602.3460.6132.002.002.003.003.003.2112.4030.6182.002.002.003.003.000.103.0340.259 294 40.019 150 01.000.501.8270.4761.002.002.002.002.003.2111.8880.4581.002.002.002.002.000.153.1910.321 317 50.029 541 01.001.00370.863359.73946.00114.75261.00512.00841.003.368500.139487.50058.00148.00353.00699.251155.000.153.1910.321 317 50.029 541 01.001.1026.12318.5859.0013.0021.0033.0050.003.36929.99321.88610.0015.0024.0039.0058.000.153.1910.321 317 50.029 541 01.001.208.1273.1875.006.007.0010.0012.003.3698.6333.3595.006.008.0010.0013.000.153.1910.321 317 50.029 541 01.001.305.0221.3554.004.005.006.007.003.3695.2651.3844.004.005.006.007.000.153.1910.321 317 50.029 541 01.001.403.8260.7873.003.004.004.005.003.3693.9940.8213.003.004.004.005.000.153.1910.321 317 50.029 541 01.000.9016.60211.4256.009.0014.0021.0031.003.36917.98912.4886.009.0015.0023.0034.000.153.1910.321 317 50.029 541 01.000.805.2352.2613.004.005.006.008.003.3695.4422.2953.004.005.007.008.000.153.1910.321 317 50.029 541 01.000.702.9941.0022.002.003.004.004.003.3693.1021.0232.002.003.004.004.000.153.1910.321 317 50.029 541 01.000.602.0910.5991.002.002.002.003.003.3692.1510.6032.002.002.002.003.000.153.1910.321 317 50.029 541 01.000.501.5920.5211.001.002.002.002.003.3691.6440.5211.001.002.002.002.00Case 4τ=20,ϕ=310.052.7220.180 379 60.009 286 71.001.00370.678353.13354.90117.00263.00514.00826.202.913500.038488.98166.00156.00350.00685.251139.100.052.7220.180 379 60.009 286 71.001.1043.46126.10718.0025.0037.0054.0077.102.91348.07029.93919.9027.0040.0061.0087.000.052.7220.180 379 60.009 286 71.001.2017.0326.18111.0013.0016.0020.0025.002.91318.1866.62411.0013.0017.0022.0027.000.052.7220.180 379 60.009 286 71.001.3011.0342.8298.009.0011.0013.0015.002.91311.6092.9388.0010.0011.0013.0016.000.052.7220.180 379 60.009 286 71.001.408.5101.7067.007.008.009.0011.002.9138.9291.7667.008.009.0010.0011.000.052.7220.180 379 60.009 286 71.000.9031.71519.58812.0018.0027.0041.0058.002.91334.43421.68413.0019.0029.0044.0063.000.052.7220.180 379 60.009 286 71.000.8011.3424.7876.008.0010.0014.0018.002.91311.9425.0176.008.0011.0015.0018.000.052.7220.180 379 60.009 286 71.000.706.5612.2544.005.006.008.0010.002.9136.8332.2964.005.006.008.0010.000.052.7220.180 379 60.009 286 71.000.604.4381.2963.004.004.005.006.002.9134.6241.3153.004.004.005.006.000.052.7220.180 379 60.009 286 71.000.503.2880.8802.003.003.004.004.002.9133.4060.8892.003.003.004.005.000.103.0470.259 130 80.019 243 31.001.00370.285362.77348.00114.00262.00508.00839.003.233499.602493.30460.00150.00348.00683.251136.000.103.0470.259 130 80.019 243 31.001.1061.12247.89717.0028.0047.0080.00123.003.23374.41160.30819.0032.0056.0099.00152.000.103.0470.259 130 80.019 243 31.001.2017.4108.3409.0012.0015.0021.0028.003.23318.9379.40810.0012.0017.0023.0031.000.103.0470.259 130 80.019 243 31.001.3010.1343.3147.008.009.0012.0015.003.23310.7953.6037.008.0010.0013.0016.000.103.0470.259 130 80.019 243 31.001.407.4291.8275.006.007.008.0010.003.2337.8221.9386.006.008.009.0010.000.103.0470.259 130 80.019 243 31.000.9032.67125.01610.0015.0026.0043.0065.003.23337.04728.01511.0017.0029.0049.0074.000.103.0470.259 130 80.019 243 31.000.8010.1395.2105.006.009.0013.0017.003.23310.7425.5075.007.0010.0013.0018.000.103.0470.259 130 80.019 243 31.000.705.5522.2473.004.005.007.008.003.2335.7862.2773.004.005.007.009.000.103.0470.259 130 80.019 243 31.000.603.6931.2662.003.003.004.005.003.2333.8091.2982.003.004.005.005.000.103.0470.259 130 80.019 243 31.000.502.6980.8122.002.003.003.004.003.2332.8010.8342.002.003.003.004.000.153.2360.321 317 50.029 541 01.001.00370.094366.48743.00110.00257.00502.00850.003.428501.415493.98061.00149.00350.00693.001155.000.153.2360.321 317 50.029 541 01.001.10102.11092.38320.0037.0075.00137.00220.003.428136.543127.62824.0047.0099.00183.00300.100.153.2360.321 317 50.029 541 01.001.2020.58112.4819.0012.0017.0026.0037.003.42823.19014.50910.0013.0019.0029.0042.000.153.2360.321 317 50.029 541 01.001.3010.4294.2206.007.009.0012.0016.003.42811.2634.6907.008.0010.0013.0017.000.153.2360.321 317 50.029 541 01.001.407.2422.1675.006.007.008.0010.003.4287.6922.3295.006.007.009.0011.000.153.2360.321 317 50.029 541 01.000.9036.26730.3549.0015.0027.0048.0076.003.42841.06934.2609.0017.0031.0055.0086.000.153.2360.321 317 50.029 541 01.000.809.9865.8774.006.009.0013.0018.003.42810.6406.3024.006.009.0013.0019.000.153.2360.321 317 50.029 541 01.000.705.1482.3213.003.005.006.008.003.4285.3992.4423.004.005.007.009.000.153.2360.321 317 50.029 541 01.000.603.3511.2612.002.003.004.005.003.4283.4801.2772.003.003.004.005.000.153.2360.321 317 50.029 541 01.000.502.4270.7932.002.002.003.003.003.4282.5200.8162.002.002.003.004.00To be specific, we considered the shift sizeΔ∈1,1.10,1.20,1.30,1.40,1.90,0.80,0.70,0.60,0.50. The shifts are considered in the percentage form, that is, the value of Δ=0.90 means that there is a 10% decrease in the parameter value, whereas the value Δ=1.10 means that the value of the parameter is increased by 10%. The value Δ=1 corresponds to no shift in the parameter value. The IC values of mean and variance of plotting statistic M, i.e., EM and VM are also given in the tables. The analysis of the proposed scheme is conducted under the following general conditions.For all of the cases given below, we haveμ=0.20Case 1.
τ=155 and ϕ=290⇒δ0=μ×ϕ=58, γ0=1−μϕ=232, τ0=155, and θ0=μ1/τ/1−μ1/τ=95.80.Case 2.
τ=96 and ϕ=148⇒δ0=μ×ϕ=29.6, γ0=1−μϕ=118.4, τ0=96, and θ0=μ1/τ/1−μ1/τ=59.15.Case 3.
τ=51 and ϕ=80⇒δ0=μ×ϕ=16, γ0=1−μϕ=64, τ0=51, and θ0=μ1/τ/1−μ1/τ=31.19.Case 4.
τ=20 and ϕ=31⇒δ0=μ×ϕ=6.2, γ0=1−μϕ=24.8, τ0=20, and θ0=μ1/τ/1−μ1/τ=11.93.
ARL0 is fixed at 370 and 500 to obtain L. The results are obtained using R software (R version 4.0.0, 2020-04-24).


### 4.1. Results and Discussion

The analysis listed in Tables3, 4 and Table S1 is conducted to assess the performance of the proposed control chart under Case 1(a). More specifically, we have considered pure shifts in shape 1 parameter of T in Table 3. For the set of parameters given in Case 1, when λe=0.05 and ARL0=370, the ARL is observed to reduce by 94.46%, 97.62%, 98.45%, and 98.82% with respect to the nominal ARL value when the shape 1 parameter of T is increased by 10%, 20%, 30%, and 40%, respectively, and reduced by 94.73%, 97.83%, 98.64%, 99.00%, and 99.20% when the shape 1 parameter of T is increased by 10%, 20%, 30%, 40%, and 50%, respectively. For the value of λe=0.10, the ARL is observed to reduce by 94.36%, 97.92%, 98.70%, and 99.03% and by 94.69%, 98.14%, 98.87%, 99.18%, and 99.37% for the same size of upward (UW) and downward (DW) shifts, respectively. Finally, for λe=0.15, the ARL is reduced by 93.86%, 98.02%, 98.82%, and 99.13% and by 94.13%, 98.25%, 98.98%, 99.28%, and 99.43%, respectively, for the same amount of the UW and DW shifts in the shape1 parameter of T.AssumingARL0=500 with λe=0.05, the ARL is reduced by 95.61%, 98.15%, 98.80%, and 99.10% when there are pure UW shifts of size 10%, 20%, 30%, and 40% in shape 1 parameter of T and reduced by 95.88%, 98.33%, 98.95%, 99.23%, and 99.39% in the case of pure DW shifts of size 10%, 20%, 30%, 40%, and 50%, respectively. A similar pattern is observed for λe=0.10 and 0.15.For the parameters given in Case 2, whenλe=0.05 and ARL0=370, the reduction in the ARL is by 90.96%, 96.45%, 97.77%, and 98.34% for 10%, 20%, 30%, and 40% for the UW shifts and by 91.50%, 96.81%, 98.07%, 98.61%, and 98.92% for 10%, 20%, 30%, 40%, and 50% DW shifts, respectively. Similarly, for λe=0.10, the ARL is reduced by 90.12%, 96.69%, 98.06%, and 98.60% and by 90.36%, 97.09%, 98.36%, 98.86%, and 99.11%, respectively, for the same amount of the UW and DW shifts. A similar pattern can be observed for λe=0.15. In the case of ARL0=500, the behavior of the ARL is observed very similar as ARL0=370.The behavior of the ARL for case 3 and case 4 can be interpreted in a similar fashion. Table4 considers pure shifts in the shape parameter of X. For the set of parameters given in case 1, when λe=0.05, the ARL is reduced by 97.24%, 98.62%, 99.01%, and 99.18% with respect to the nominal value of ARL0=370 when Δτ∈1.10,1.20,1.30,1.40, respectively. Furthermore, when Δτ∈0.90,0.80,0.70,0.60,0.50, the ARL is reduced by 97.71%, 98.98%, 99.35%, 99.49%, and 99.64%, respectively. For λe=0.10 and 0.15, a similar reduction in the ARL is observed for both 370 and 500 nominal values of the ARL. Similarly, for case 2 of the parameters, the ARL is reduced by 97.54%, 98.86%, 99.18%, and 99.36% when there is the UW shift of the size mentioned above and by 98.00%, 99.17%, 99.46%, 99.61%, and 99.72% when there is a DW shift of similar size as mentioned above. Finally, for λe=0.15, the ARL is reduced by 97.59%, 98.97%, 99.28%, and 99.43% for the UW shifts and by 98.10%, 99.25%, 99.51%, 99.68%, and 99.73% for the DW shifts in the shape parameter of X. The interpretation of the remaining cases can be done similarly. According to the ARL criteria, the chart based on pure shift in the shape parameter of X performs better than the one based on pure shift in shape 1 parameter of T.For simultaneous shifts in the two parameters of X and T, TableS1, the similar degenerating pattern of ARL is observed for increasing UW and DW shifts. Specifically, when Δτ=1.10 and Δδ∈1.10,1.20,1.30,1.40 in Case 1, i.e., simultaneous 10% UW shift in shape parameter of X and 10%, 20%, 30%, and 40% UW shift in the shape 1 parameter of T, the ARL is reduced by 97.35%, 97.90%, 98.47%, and 98.83%, respectively. On the other hand, when Δδ∈0.90,0.80,0.70,0.60,0.50, the ARL is reduced by 97.36%, 98.00%, 98.65%, 99.00%, and 99.20% for the same amount of shift in the shape parameter of X. In addition, when a UW shift in the shape parameter of X is increased to 20%, the ARL is reduced by 98.62%, 98.64%, 98.73%, and 98.89% and by 98.62%, 98.64%, 98.77%, 99.01%, and 99.20% for similar UW and DW shifts in the shape 1 parameter of T as given above. The same degeneration in the ARL values is observed for various values of smoothing parameter. However, it is worth noting that the proposed chart performs exceptionally well for the Case 1 and Case 2 of the parameters, but as we move down the table to Case 3 and Case 4, the performance deteriorated, especially in the case of smaller UW and DW shifts whether pure or simultaneous. For example, when λe=0.15 and ARL0=500 in Case 4 of Table 3, the ARL reduced only by 66.62% for 10% UW and only by 31.21% for 10% DW shift which is substantially lower compared to Case 1 for similar value of ARL0 and λe. When there is a simultaneous shift of size Δδ,Δτ=1.10,1.10 in Case 4 of Table S1, for ARL0=370 and λe=0.15, the ARL reduced only by 81.02%, whereas in Case 1, the reduction in ARL is by 97.76% for a similar shift size and smoothing parameter. The standard deviation of ARL (SDRL) has the same decreasing pattern for all of the cases. Thus, it is concluded that the ARL and SDRL decrease by increasing the UW or DW shifts.The percentile analysis shows that, in most cases, the ARL is greater than the median (q50) but less than the75th percentile (q75) suggesting that the run length distribution is positively skewed. For example, when Δδ=1.10, λe=0.05, and ARL0=370 in Case 1 of Table 3, the ARL is 20.497, the median is 18, and q75 is 25. Interestingly, in some cases the ARL is slightly less than the median which means that the distribution is slightly negatively skewed. For example, in case 1 of Table 3 assuming Δδ=1.30 and λe=0.05, the ARL is 5.741 and the median is 6; however, in Table 4, when Δτ=1.30 and λe=0.05 in case 1, the ARL is 3.675 and the median is 4. Thus, the proposed chart is more sensitive to small shifts Δδ,Δτ∈1.10,0.90 for pure or simultaneous shifts with λe=0.05. It is to be noted that the sensitivity to moderate shifts increased when λe increased. For example, in case 1 of Table 3, when Δδ=1.10,0.90 (small shifts) and λe=0.05, the ARL1 is 20.497 and 19.524, while the corresponding ARL1 values with λe=0.15 are 22.761 and 21.766, respectively. On the other hand, when Δδ=1.40 (a large shift), the ARL1 value for λe∈0.05,0.10,0.15 is (4.354,3.588,3.213), respectively. For a simultaneous shift of size Δδ,Δτ=1.10,1.10 as reported in case 4 in Table S1, the ARL1 is 38.66 for λe=0.05 and 70.28 for λe=0.15. However, this does not hold for all cases. For example, in Case 1 of Table S1 when λe∈0.05,0.15 the ARL1 is (9.83,8.28) for 10% simultaneous UW shift in both parameters. It is worth noting that the behavior of the ARL does not conform to the traditional fashion, that is, small smoothing parameters are suitable for small shifts and vice versa. As this discrepancy exists only for a few ARL1 values, it is safe to say that smaller values of λe are suitable for detecting smaller shifts and larger values are suitable to detect larger shifts.In TableS2, we have considered pure shift in the shape 2 parameter of T, whereas in Table S3, pure shift in the rate parameter of X is considered. Finally, a simultaneous shift in shape 2 parameter of T and rate parameter of X is considered in Table S4. From the tables, we observed a very similar degenerating pattern in the ARL as well as in the SDRL as noted earlier. More specifically, in the case of pure shift, we can see that an UW shift in the shape 1 parameter of T is detected more quickly as compared to the UW shift in the shape 2 parameter of T. On the other hand, a DW shift in shape 2 is detected more quickly. For example, when λe=0.05, ARL0=370, and Δδ∈1.30,1.40 in Case 1 of Table 3, ARL1=5.741,4.354, and when Δγ=1.30,1.40 under similar settings in Table 4, ARL1=6.735,5.278. On the other hand, when Δδ∈0.90,0.80 in Case 1, ARL1=19.524,8.024 and ARL1=17.912,6.971 when Δγ=0.90,0.80 in Case 1. While comparing the efficiency of the proposed chart for detecting pure shifts in shape and rate parameter of X, we can see that an UW shift in the rate parameter is detected more quickly, whereas a DW shift in the shape parameter of X is detected more quickly. The performance of the chart for simultaneous shifts in Table S1 is more or less the same as given in Table S4.


## 4.1. Results and Discussion

The analysis listed in Tables3, 4 and Table S1 is conducted to assess the performance of the proposed control chart under Case 1(a). More specifically, we have considered pure shifts in shape 1 parameter of T in Table 3. For the set of parameters given in Case 1, when λe=0.05 and ARL0=370, the ARL is observed to reduce by 94.46%, 97.62%, 98.45%, and 98.82% with respect to the nominal ARL value when the shape 1 parameter of T is increased by 10%, 20%, 30%, and 40%, respectively, and reduced by 94.73%, 97.83%, 98.64%, 99.00%, and 99.20% when the shape 1 parameter of T is increased by 10%, 20%, 30%, 40%, and 50%, respectively. For the value of λe=0.10, the ARL is observed to reduce by 94.36%, 97.92%, 98.70%, and 99.03% and by 94.69%, 98.14%, 98.87%, 99.18%, and 99.37% for the same size of upward (UW) and downward (DW) shifts, respectively. Finally, for λe=0.15, the ARL is reduced by 93.86%, 98.02%, 98.82%, and 99.13% and by 94.13%, 98.25%, 98.98%, 99.28%, and 99.43%, respectively, for the same amount of the UW and DW shifts in the shape1 parameter of T.AssumingARL0=500 with λe=0.05, the ARL is reduced by 95.61%, 98.15%, 98.80%, and 99.10% when there are pure UW shifts of size 10%, 20%, 30%, and 40% in shape 1 parameter of T and reduced by 95.88%, 98.33%, 98.95%, 99.23%, and 99.39% in the case of pure DW shifts of size 10%, 20%, 30%, 40%, and 50%, respectively. A similar pattern is observed for λe=0.10 and 0.15.For the parameters given in Case 2, whenλe=0.05 and ARL0=370, the reduction in the ARL is by 90.96%, 96.45%, 97.77%, and 98.34% for 10%, 20%, 30%, and 40% for the UW shifts and by 91.50%, 96.81%, 98.07%, 98.61%, and 98.92% for 10%, 20%, 30%, 40%, and 50% DW shifts, respectively. Similarly, for λe=0.10, the ARL is reduced by 90.12%, 96.69%, 98.06%, and 98.60% and by 90.36%, 97.09%, 98.36%, 98.86%, and 99.11%, respectively, for the same amount of the UW and DW shifts. A similar pattern can be observed for λe=0.15. In the case of ARL0=500, the behavior of the ARL is observed very similar as ARL0=370.The behavior of the ARL for case 3 and case 4 can be interpreted in a similar fashion. Table4 considers pure shifts in the shape parameter of X. For the set of parameters given in case 1, when λe=0.05, the ARL is reduced by 97.24%, 98.62%, 99.01%, and 99.18% with respect to the nominal value of ARL0=370 when Δτ∈1.10,1.20,1.30,1.40, respectively. Furthermore, when Δτ∈0.90,0.80,0.70,0.60,0.50, the ARL is reduced by 97.71%, 98.98%, 99.35%, 99.49%, and 99.64%, respectively. For λe=0.10 and 0.15, a similar reduction in the ARL is observed for both 370 and 500 nominal values of the ARL. Similarly, for case 2 of the parameters, the ARL is reduced by 97.54%, 98.86%, 99.18%, and 99.36% when there is the UW shift of the size mentioned above and by 98.00%, 99.17%, 99.46%, 99.61%, and 99.72% when there is a DW shift of similar size as mentioned above. Finally, for λe=0.15, the ARL is reduced by 97.59%, 98.97%, 99.28%, and 99.43% for the UW shifts and by 98.10%, 99.25%, 99.51%, 99.68%, and 99.73% for the DW shifts in the shape parameter of X. The interpretation of the remaining cases can be done similarly. According to the ARL criteria, the chart based on pure shift in the shape parameter of X performs better than the one based on pure shift in shape 1 parameter of T.For simultaneous shifts in the two parameters of X and T, TableS1, the similar degenerating pattern of ARL is observed for increasing UW and DW shifts. Specifically, when Δτ=1.10 and Δδ∈1.10,1.20,1.30,1.40 in Case 1, i.e., simultaneous 10% UW shift in shape parameter of X and 10%, 20%, 30%, and 40% UW shift in the shape 1 parameter of T, the ARL is reduced by 97.35%, 97.90%, 98.47%, and 98.83%, respectively. On the other hand, when Δδ∈0.90,0.80,0.70,0.60,0.50, the ARL is reduced by 97.36%, 98.00%, 98.65%, 99.00%, and 99.20% for the same amount of shift in the shape parameter of X. In addition, when a UW shift in the shape parameter of X is increased to 20%, the ARL is reduced by 98.62%, 98.64%, 98.73%, and 98.89% and by 98.62%, 98.64%, 98.77%, 99.01%, and 99.20% for similar UW and DW shifts in the shape 1 parameter of T as given above. The same degeneration in the ARL values is observed for various values of smoothing parameter. However, it is worth noting that the proposed chart performs exceptionally well for the Case 1 and Case 2 of the parameters, but as we move down the table to Case 3 and Case 4, the performance deteriorated, especially in the case of smaller UW and DW shifts whether pure or simultaneous. For example, when λe=0.15 and ARL0=500 in Case 4 of Table 3, the ARL reduced only by 66.62% for 10% UW and only by 31.21% for 10% DW shift which is substantially lower compared to Case 1 for similar value of ARL0 and λe. When there is a simultaneous shift of size Δδ,Δτ=1.10,1.10 in Case 4 of Table S1, for ARL0=370 and λe=0.15, the ARL reduced only by 81.02%, whereas in Case 1, the reduction in ARL is by 97.76% for a similar shift size and smoothing parameter. The standard deviation of ARL (SDRL) has the same decreasing pattern for all of the cases. Thus, it is concluded that the ARL and SDRL decrease by increasing the UW or DW shifts.The percentile analysis shows that, in most cases, the ARL is greater than the median (q50) but less than the75th percentile (q75) suggesting that the run length distribution is positively skewed. For example, when Δδ=1.10, λe=0.05, and ARL0=370 in Case 1 of Table 3, the ARL is 20.497, the median is 18, and q75 is 25. Interestingly, in some cases the ARL is slightly less than the median which means that the distribution is slightly negatively skewed. For example, in case 1 of Table 3 assuming Δδ=1.30 and λe=0.05, the ARL is 5.741 and the median is 6; however, in Table 4, when Δτ=1.30 and λe=0.05 in case 1, the ARL is 3.675 and the median is 4. Thus, the proposed chart is more sensitive to small shifts Δδ,Δτ∈1.10,0.90 for pure or simultaneous shifts with λe=0.05. It is to be noted that the sensitivity to moderate shifts increased when λe increased. For example, in case 1 of Table 3, when Δδ=1.10,0.90 (small shifts) and λe=0.05, the ARL1 is 20.497 and 19.524, while the corresponding ARL1 values with λe=0.15 are 22.761 and 21.766, respectively. On the other hand, when Δδ=1.40 (a large shift), the ARL1 value for λe∈0.05,0.10,0.15 is (4.354,3.588,3.213), respectively. For a simultaneous shift of size Δδ,Δτ=1.10,1.10 as reported in case 4 in Table S1, the ARL1 is 38.66 for λe=0.05 and 70.28 for λe=0.15. However, this does not hold for all cases. For example, in Case 1 of Table S1 when λe∈0.05,0.15 the ARL1 is (9.83,8.28) for 10% simultaneous UW shift in both parameters. It is worth noting that the behavior of the ARL does not conform to the traditional fashion, that is, small smoothing parameters are suitable for small shifts and vice versa. As this discrepancy exists only for a few ARL1 values, it is safe to say that smaller values of λe are suitable for detecting smaller shifts and larger values are suitable to detect larger shifts.In TableS2, we have considered pure shift in the shape 2 parameter of T, whereas in Table S3, pure shift in the rate parameter of X is considered. Finally, a simultaneous shift in shape 2 parameter of T and rate parameter of X is considered in Table S4. From the tables, we observed a very similar degenerating pattern in the ARL as well as in the SDRL as noted earlier. More specifically, in the case of pure shift, we can see that an UW shift in the shape 1 parameter of T is detected more quickly as compared to the UW shift in the shape 2 parameter of T. On the other hand, a DW shift in shape 2 is detected more quickly. For example, when λe=0.05, ARL0=370, and Δδ∈1.30,1.40 in Case 1 of Table 3, ARL1=5.741,4.354, and when Δγ=1.30,1.40 under similar settings in Table 4, ARL1=6.735,5.278. On the other hand, when Δδ∈0.90,0.80 in Case 1, ARL1=19.524,8.024 and ARL1=17.912,6.971 when Δγ=0.90,0.80 in Case 1. While comparing the efficiency of the proposed chart for detecting pure shifts in shape and rate parameter of X, we can see that an UW shift in the rate parameter is detected more quickly, whereas a DW shift in the shape parameter of X is detected more quickly. The performance of the chart for simultaneous shifts in Table S1 is more or less the same as given in Table S4.


## 5. Real Data Application

In this section, the proposed methodology is applied to a real data set taken from a disaster management department that controls fire [20]. Controlling fire in an efficient manner leaves a positive impact than an uncontrolled occurrence of fire that may cause severe destruction. A disaster management department always tries to monitor the damages inflicted by the fire outbreaks to minimize the future losses. There are 25 observations in the data set including two variables: the amount of losses (in 1000$) and the time interval (in days) between the successive fire outbreaks. Both of these variables are transformed into proportion. The magnitude (X) represents the proportion of losses, whereas the TBE (T) represents the proportion of days between the successive fire outbreaks. Our interest here is to detect an increase in the amount of losses and/or decrease in the time between the fire outbreaks. The variable T is fitted to beta distribution, and variable X is fitted to unit gamma distribution. To estimate the parameters, we have used the maximum likelihood approach, and all the calculations are done using the statistical software R. Apart from the first 25 observations which are used as the phase I sample, we have further simulated 25 values as the phase II observations by introducing shifts of Δδ=0.80 and Δτ=0.80 to exhibit the application of the proposed charting scheme. Empirical mean and variance of the plotting statistic are 0.254 5 and 0.021 4, respectively. Using λe=0.15andL=1.90, we obtained UCL=0.5325 with ARL0=25. The maximum likelihood estimates of the parameters along with their standard errors (in parentheses), the Akaike information criterion (AIC) and Bayesian information criterion (BIC), for the 25 IC observations are presented in Table 5. Figure 1 presents the Max-EWMA control chart constructed for the data set (phase I and phase II). There appears no signal in phase I data as the chart gives no OOC signal; however, the first OOC signal is observed at the 28th sample. When a shift in the fire outbreaks, such as this one, is observed, the disaster management department can look into the reasons behind the phenomenon and try to eliminate them.Table 5
Parameter estimates with standard errors (in parentheses) and goodness-of-fit statistics for the proportion of days between successive fire outbreaks and proportion of losses.
DistributionML estimatesAICBICBetaμ^0.040 2(0.006 7)−106.889 3−104.451 6ϕ^21.108 9(2.099 4)Unit gammaμ^0.922 2(0.019 7)−124.597 7−122.160 0τ^39.733 8(10.842 3)Figure 1
Max-EWMA chart for monitoring the fire outbreaks considering simultaneous shift in shape 1 parameter of beta distribution and shape parameter of unit gamma distribution.Furthermore, we have constructed the Max-EWMA chart for 80% pure shift in the parameters. Specifically, when there is a pure shift in the shape 1 parameter of T, Figure2 shows that the first OOC signal is triggered at the 34th sample. On the other hand, if there is a pure shift in the shape parameter of the unit gamma distribution, Figure 3, the first OOC signal is triggered at the 30th sample. These charts indicate that the simultaneous shift is detected more quickly as compared to the pure shift. The OOC points are labelled with respective labels to identify the signals.Figure 2
Max-EWMA chart for monitoring the fire outbreaks considering the pure shift in the shape 1 parameter of beta distribution.Figure 3
Max-EWMA chart for monitoring the fire outbreaks considering the pure shift in the shape parameter of unit gamma distribution.


## 6. Conclusion

With the advancement in technology, many human centric jobs have been taken over by machines and it is imperative to monitor the output of these machines to assess whether the performance level is up to the expectations or not. The SPC provides tools and techniques which enabled us to distinguish between assignable and natural causes in a process. A number of techniques have already been developed to monitor processes with different assumptions, distributional forms, and range of data involved; however, there are very few studies related to unit interval data. Therefore, motivated by the importance of unit interval data, we introduced a Max-EWMA chart for simultaneous monitoring of the magnitude and TBE of an event under the assumption that TBE follows the beta distribution and magnitude of the event follows unit gamma distribution. In order to evaluate the performance of the proposed chart, the most frequently used criterion, i.e., the ARL has been used. The results indicate that the overall performance of the proposed chart is substantially good for small to medium-sized shifts. The current study also emphasizes on the fact that simultaneous monitoring of magnitude and frequency is more efficient as compared to individual monitoring of both these characteristics. Finally, the charting procedure is applied to a real life data set. In the future, the current work can be extended by considering some new distributions dealing with unit interval data. Furthermore, the effect of parameter estimation on phase I and phase II may be studied in detail. Also, a recent idea of neutrosophic statistics [28] can be utilized to extend the present work.



---

*Source: 7951748-2022-02-21.xml*