Max-EWMA Chart Using Beta and Unit Nadarajah and Haghighi Distributions

Te recent industrial revolution is a result of modern technological advancement and industrial improvements require quick detection of assignable causes in a process. Tis study presents a monitoring scheme for unit interval data assuming beta and unit Nadarajah and Haghighi distributions. To this end, a maximum exponentially weighted moving average (Max-EWMA) chart is introduced to jointly monitor unit interval bounded time and magnitude data. Te performance of the proposed chart is evaluated by using average run length and other characteristics of run length distribution using extensive Monte Carlo simulations. Besides a comprehensive simulation study, a real data set is also used to assess the performance of the chart. Te results supplementing the proposed chart are efcient for joint monitoring time and magnitude, and simultaneous shifts are detected more quickly than separate shifts in the process parameters.


Introduction
In the context of services or goods provided by industry, quality excellence plays an important role that needs to be judged by a metric. If a good or service is termed to be of superior quality, there must be a standard to determine which products or services are of superior quality. Tus, there arises a need to establish specifcations, measurement systems, acceptability standards, and methods for the result assessment. To improve monitoring and control, statistical quality control (SQC) tools are widely used. A fundamental tool of SQC is the control chart, which is a graphical presentation of the process history. If a monitoring statistic falls within the boundaries of a control chart, it is termed as incontrol (IC); otherwise, it is termed as out-of-control (OOC).
Tere are two types of control charts. Te frst one is the memory-less charts that use the information of the current sample only while completely ignoring the process history.
Tese charts work well with a large shift in the process. Attribute charts like p and np using Bernoulli distribution are commonly used to monitor the fraction of the number of nonconforming items [1,2]. However, due to the memoryless nature of these charts, the historical information of the process is ignored as the new points are plotted [3]. As a result, these charts cannot detect small shifts in process parameters satisfactorily. Consequently, memory-type control charts such as the exponentially weighted moving average chart (EWMA) and the cumulative sum (CUSUM) chart are required. Te EWMA chart is more sensitive as compared to the Shewhart chart because it takes into account the historical information of the process. Te EWMA chart is a Markov process and simpler than a CUSUM chart. It is reported in the literature that the EWMA chart is not sensitive to the non-normality of data for small smoothing values [4].
Many processes require very high precision, also known as the high quality processes, which cannot be monitored by using ordinary attribute control charts. To this end, timebetween events (TBE) control charts are used for a high quality process monitoring, for example, the production process of an armed facility requires utmost precision. Te recent popularity of TBE charts is an indication that these charts are suitable for high quality processes such as high yield production lines with extremely small defect rates [5] because these charts overcome the drawbacks of the traditional Shewhart charting schemes [6]. Ali [7] proposed a TBE control chart based on the renewal process assuming a continuous exponentiated family of distributions. Some recent works on TBE can be seen in [8], [9], [10,11], [12,13], and [14].
In many cases, the impact of an event occurrence on the process performance is determined by the event magnitude or frequency. Tus, the occurrence of events along their magnitudes needs to be monitored simultaneously with sophisticated tools. Wu et al. [15] developed a rate chart for monitoring time and magnitude simultaneously by using the ratio of magnitude and time. In another study, Wu et al. [16] utilized a Shewhart control chart for monitoring the frequency and magnitude of an event simultaneously. Some recent contributions in this direction can be seen in [17], [18], [19], and references cited therein.
In many practical settings, our interest lies in the monitoring processes bounded in unit interval (0, 1) and existing charts such as based on Bernoulli distribution [20] have poor performance. Tus, there is a dire need to develop control charts assuming other suitable distributions to monitor rates or proportion data. Many practical situations where such data are frequently encountered include unemployment rate, TV rating, and area (in proportion) specifed for agricultural purposes [21]. Sant'Anna and Caten [22] used beta distribution to construct a Shewhart control chart for monitoring fractional type data. Lima-Filho et al. [23] introduced an infated beta control chart. Simplex distribution is somehow similar to exponential dispersion family as well as normal distribution; that is, if the dispersion parameter of simplex distribution is small, this distribution is similar to the normal distribution and becomes fexible to cover many features of data if the dispersion parameter is large [24]. Tis distribution also has applications in medical science such as mapping the proportion of remaining gas volume relative to its initial volume in the eyes of patients [21,25]. Zhang and Qiu [26] presented a regression analysis for proportion data using the simplex distribution.
To monitor time and magnitude data bounded in the (0, 1) interval, this article proposes a Max-EWMA chart. Te plotting statistic of this chart comprises the maximum the absolute of two independent EWMA statistics. Tus, it is suitable for simultaneously monitoring shifts in magnitude (X) as well as TBE (T) of events. Te performance of the chart is assessed by using diferent average run length (ARL) characteristics. Besides comprehensive Monte Carlo simulations, a real data example is also part of the study to show the application of the proposed method.
Te rest of the study is organized as follows. Section 2 presents information regarding beta and unit Nadarajah and Haghighi distributions. Te construction of the proposed chart for time and magnitude monitoring is also discussed in the same section. In-and out-of-control run length properties are discussed in Section 3. Real life application of the proposed chart is presented in Section 4. Finally, the concluding remarks are discussed in Section 5.

Max-EWMA Control Chart to Monitor Unit Bounded Magnitude and Time Data
To monitor the processes based on unit interval bound variables, a Max-EWMA control chart is proposed. To design the Max-EWMA chart, we used beta and unit Nadarajah and Haghighi distributions.

Unit Nadarajah and Haghighi Distribution.
Te UNH distribution is proposed by Shah et al. [27]. Te probability density function of Nadarajah and Haghighi (NH) distribution is where α > 0 and λ > 0 are the shape and rate parameters, respectively. Using the transformation T � exp(− Y), one can obtain the unit NH density as 2

Journal of Mathematics
Te mean and variance of this distribution can be computed numerically.

Max-EWMA Chart.
In order to develop the proposed charting scheme, assume that the TBE follows UNH distribution, T ∼ UNH(α, λ), and magnitude follows beta distribution, X ∼ B(c, δ). Furthermore, the process is assumed to start at t � 0, and IC parameters are denoted by (α 0 , λ 0 ) and (c 0 , δ 0 ). For the sake of simplifcation, independence between the magnitude and TBE is assumed. Te shifts in the parameters of X and T, respectively, are considered as follows. c � Δ c c 0 , δ � Δ δ δ 0 , α � Δ α α 0 , and λ � Δ λ λ 0 . Here, Δ α , Δ λ , Δ c , and Δ δ represent the shift in the shape and rate parameters of UNH distribution and shape 1 and shape 2 parameters of the beta distribution, respectively. Te process is IC when both Δ λ and Δ c are equal to one, i.e., Δ λ � Δ c � 1. For constructing the Max-EWMA control chart, the standardized statistics are defned as follows: where μ UNH and σ 2 UNH represent the mean and variance of the UNH which are computed numerically. Using these statistics, two independent EWMA statistics are constructed as where θ e ∈ (0, 1] is known as the smoothing constant of the Max-EWMA control chart, and L is the constant multiplier selected to have a desired IC run length. In the literature, a large value of the smoothing parameter is suitable for detecting a large shift and vice versa. Typically, to detect small to medium size shifts, we use θ e ∈ [0.05, 0.25] [28]. Te initial values of the EWMA statistics are fxed at 0, i.e., Y 0 � E(U k ) � 0 and Z 0 � E(V k ) � 0. Te charting statistic of the proposed chart is given by where

Run Length Properties of the Proposed Chart
Te average run length (ARL) is widely used as a performance assessment measure of a control chart. It is defned as the mean value of the number of samples or subgroups that are plotted until an OOC signal occurs. Te IC ARL is calculated assuming that the process is operating under natural variations and denoted by ARL 0 , while the OOC ARL is calculated when the process is operating under a shift and denoted by ARL 1 . If there is no shift in the process, no OOC signal should occur, but due to the natural variations in a process, a false alarm signal does occur. On the other hand, when the process moves to an OOC state, an immediate OOC signal should be detected by a chart and adjustments should be made immediately. Tus, it is required that ARL 0 should be large so that unnecessary inspection of the process is avoided and ARL 1 should be small to quickly adjust the process. A smaller ARL 1 value has become a standard of efciency when comparing the performance of diferent control charts. To study the properties of the run length (RL)  More specifcally, we considered UW shifts as Δ ∈ (1.40, 1.30, 1.20, 1.10) and DW shifts as Δ ∈ (0.50, 0.60, 0.70, 0.80, 0.90), where Δ � 1 indicates that the respective parameter is IC. Te value Δ � 0.50 indicates a DW shift of a 50% size, and Δ � 1.40 represents an UW shift of size 40%. Te IC values of mean and variance of the monitoring statistic, denoted by EM and VM, are also listed in the tables. Te cases considered for the analysis are as follows:  Table  4: ARL results of the Max-EWMA control chart for a pure shift in the shape 1 parameter of beta distribution (Δ 500), and θ e ∈ (0.05, 0.10, 0.15).    Journal of Mathematics 9  μ 0 � 0.20 is considered for all of the cases given above. Te IC values of the parameters of UNH distribution are λ 0 � 1 and α 0 ∈ (0.50, 1.50). Furthermore, the IC ARL is fxed at ARL 0 ∈ (370, 500). Te results are obtained using an algorithm written in statistical software R (version 4.0.0 (2020-04-24)).

Results and Discussion.
A detailed discussion on the performance of the proposed Max-EWMA control chart is given in this section. Considering a pure shift in the shape 1 parameter of beta distribution with α � 0.50 and ARL 0 � 370 in Table 4 It is noticed that the performance of the control chart under a pure shift in the rate parameter of T is poor as many ARL 1 values are either large or biased (ARL 0 < ARL 1 ). On the other hand, in the case of a shift in the frst shape parameter of X, the proposed chart performs much better compared to the former. However, as our purpose lies in the joint monitoring, the proposed chart performs better for joint shifts as compared to a pure shift in the rate parameter of T, especially in Case 1 and Case 2. Te SDRL, similar to the ARL, decreased by increasing the shift size. Te percentile analysis shows that the ARL has a rightly skewed distribution because the ARL is greater than the 50 th percentile (median) and less than 75 th percentile.
In addition, the proposed chart shows a high sensitivity for smaller shifts and smoothing parameter values. However, its sensitivity, for moderate to large shifts, increases as θ e increases. For example, in Case 1 of Table 4 with Δ c � 0.90, ARL 1 � 19.242 when θ e � 0.05 and ARL 1 � 21.465 for θ e � 0.15. Similarly, in Case 2 of Table S1 with (Δ c , Δ λ ) � (1.10, 1.10), ARL 1 � 32.486 for θ e � 0.05 and ARL 1 � 40.554 for θ e � 0.15. Tus, it is concluded that the smaller values of θ e are suitable for detecting smaller shifts and vice versa [28].
Tables S2-S4 are tabulated to study the shift in shape 2 parameter of beta distribution, shape parameter of UNH distribution, and simultaneous shift in these two parameters, respectively. Te proposed chart behaves very similarly as discussed previously. Te chart is efcient in detecting pure shifts in the shape 2 parameter of X and inefcient in detecting pure shifts in the shape parameter of T. Te performance of the chart with simultaneous shifts is more or less the same. However, one can notice that a pure UW shift in the shape 1 parameter of X is detected more quickly as compared to a pure UW in the shape 2 parameter of X and vice versa. Similarly, simultaneous UW shifts in the shape 1 parameter of X and rate parameter of T are detected quickly as compared to the simultaneous UW shifts in the shape 2 parameter of X and shape parameter of T and vice versa. Te chart performs poorly for pure shifts in either parameter of UNH distribution. Te interpretation of the rest of the tables from Table S5 to Table S10, which are constructed assuming α � 1.50, can be done on similar lines.

A Real Data Application
Tis section shows the implementation of the proposed chart on a real-life data set. Te data set consists of shooting percentages of 292 players as reported by Simonof [29]. Tis data set is considered as the magnitude (X). As the relevant TBE data set is not available, we generated TBE data (T) assuming the UNH distribution with the IC parameters α 0 � 0.451 and λ 0 � 0.95. Te TBE data set represents the proportion of minutes spent on the training by the players (out of 120 minutes) daily. Te beta distribution is ftted to X and UNH on T, whereas the estimates of the parameters are obtained by using the maximum likelihood method. Tese 292 observations of X and T are assumed to be IC and are used as Phase I data, whereas 58 OC observations are generated as the Phase II data by introducing 15% UW shift in the shape 1 parameter of X and shape parameter of T, i.e., Δ c � Δ λ � 1.15. We considered the following parameters to construct the chart θ e � 0.10, L � 2.671, UCL � 0.289, and ARL 0 � 292.
Te estimates of the parameters of Phase I data through the maximum likelihood method are listed in Table 6. Besides these estimates, this table also includes standard errors (given in parentheses) of these estimates, i.e., AIC and BIC. In Figure 1, a control chart is constructed for the combined Phase I and Phase II data set. As all of the Phase I data are plotted under the UCL, the chart triggers the OOC signal for the frst time at the 305 th sample. Te reason behind the occurrence of the OOC signal is the presence of an assignable cause in the shooting percentage.

Conclusion
Generally, a process deteriorates due to assignable causes and we need efcient monitoring tools to detect such changes. Tere are very few charts for monitoring unit interval bounded data. To this end, we modifed the EWMA chart and introduced a Max-EWMA control chart for monitoring TBE and magnitude data simultaneously. As the charting statistic of the proposed chart is comprised of two independent EWMA statistics, the proposed chart can detect small to medium UW and DW shifts efectively. We have considered diferent shifts and values of the smoothing parameter to assess the behavior of the control charts. Based on the beta and unit Nadarajah and Haghighi distributions, the efciency of the proposed chart is evaluated in terms of diferent run length characteristics. Te results indicated that the proposed chart is efcient for small or medium size simultaneous shifts. Te performance of the chart is observed to be poor in the case of pure shifts suggesting that charts based on simultaneous shifts are more efcient as compared to pure shifts. Finally, a real data example is also shown to illustrate the implementation the chart in practice. In the future, the efect of estimation on in-and out-of-control ARL can be studied. Also, the use of the smoothing parameter value based on the fact that the smaller value is suitable for smaller shifts and vice versa is not very reliable, since, in real life applications, we cannot predict whether a shift size is large or small. To address this particular issue, an adaptive Max-EWMA chart can be used. Furthermore, other unit interval distributions, including truncated normal, Kumarshwamy, unit logistic, log-Lindley, unit Birnbaum-Saunders, and unit Gompertz, can also be used to construct time and magnitude charts.

Data Availability
Te data used in the study can be requested from the corresponding author.