The exponentiated exponentially weighted moving average control chart

Abstract

Memory-type control charts are widely used for monitoring small to moderate shifts in the process parameter(s). In the present article, we present an exponentiated exponentially weighted moving average (Exp-EWMA) control chart that weights the past observations of a process using an exponentiated function. We evaluated the run-length characteristics of the Exp-EWMA chart via Monte Carlo simulations. A comparison study versus the CUSUM, EWMA and extended EWMA (EEWMA) charts under similar in-control (IC) run-length properties demonstrates that the Exp-EWMA chart is more effective for detecting small and, under certain circumstances, moderate shifts for both the zero-state and steady-state cases. Moreover, the Exp-EWMA chart has better zero-state out-of-control (OOC) performance than an EWMA chart with smoothing parameter equal to the limit to the infinity of the exponentiated function, while the two charts perform similarly for the steady-state case. Finally, it is shown that the Exp-EWMA chart is more IC robust than its competitors under several non-normal distributions. Two examples are provided to explain the implementation of the proposed chart

1 Introduction

Control charts are the most important tool of Statistical Process Control (SPC) and are used in many manufacturing and non-manufacturing processes to detect shifts in the quality characteristic of interest. The process variations are classified into two categories; the common-cause of variation and the assignable-cause of variation. A process operating with only common-cause of variation is considered to be in-control (IC), while a process operating with assignable-cause of variation is said to be out-of-control (OOC). There are two types of control charts in SPC literature; the memory-less and the memory-type charts. The memory-less charts were first introduced by Shewhart (1926). These charts are easy to use and effective in detecting large shifts in the process parameter(s). However, they do not have good detection ability for small and moderate shifts because their charting statistics are based only on the current observation. The most popular memory-type control charts are the cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) charts, proposed by Page (1954) and Roberts (1959) respectively, where their charting statistics are based on both the current and past observations. These charts are very sensitive in detecting small and moderate shifts.

The properties of the EWMA chart have been studied by many authors. Crowder (1987) evaluated the average run-length (ARL) values of an EWMA chart with asymptotic control limits using integral equations, while later, Crowder (1989) investigated the design of the EWMA chart. Saccucci and Lucas (1990) presented a computer program for evaluating the zero-state and steady-state ARL values of the EWMA chart. Lucas and Saccucci (1990) studied the properties of an EWMA chart with asymptotic control limits and they also presented the optimal EWMA schemes for specific sizes of shifts. Moreover, they presented three enhancements of it; those are (i) the EWMA chart with fast initial response (FIR) feature that makes it more effective when the shift occurs at the beginning, (ii) the combined Shewhart-EWMA chart which is very effective for both small and large shifts and (iii) the robust EWMA chart that provides protection against outliers. Chandrasekaran et al. (1995) proposed a stochastic model of the EWMA chart with variance-adjusted control limits. Steiner (1999) evaluated the run-length properties of the EWMA chart with time-varying control limits using non-homogeneous Markov chains and he concluded that using time-varying control limits is closely related to the FIR feature and provides good performance to early process shifts. Knoth (2005) compared the performance of several EWMA schemes with FIR feature. Human et al. (2011) investigated the IC robustness of the EWMA chart with asymptotic control limits and they showed that an EWMA chart is robust to several non-normal distributions for small values of smoothing parameters. Abbas et al. (2014) presented an EWMA chart using auxiliary information and they showed that it performs better than the univariate and bivariate EWMA charts for small and moderate shifts. Knoth (2015) computed the run-length quantiles for several EWMA schemes to compare their zero-state and steady-state performances.

Several extensions and modifications of the EWMA chart have been proposed in the SPC literature to improve the detection ability of the traditional EWMA chart for specific ranges of shifts. Shamma and Shamma (1992) proposed the double EWMA (DEWMA) chart to make the EWMA chart more efficient for small shifts. Capizzi and Masarotto (2003) presented the adaptive EWMA (AEWMA) chart where the past observations are weighted using a function of the current error. Sheu and Lin (2003) introduced the generally weighted moving average (GWMA) chart which is a generalized extension model of the EWMA chart. Patel and Divecha (2011) presented the modified EWMA (MEWMA) chart which is an enhancement of the EWMA chart. Abbas et al. (2013) proposed the mixed EWMA-CUSUM (MEC) chart which is more effective than the EWMA and CUSUM charts for small shifts. Abbas et al. (2013) presented the progressive mean (PM) chart which is a cumulative average of all observations. Khan et al. (2017) studied in more detail the MEWMA chart and they showed that it is more effective than the classical EWMA chart. Naveed et al. (2018) presented an extended EWMA (EEWMA) chart which gives a positive weight to the current observation and a negative weight to the preceding observations. The EEWMA chart was found more sensitive in detecting shifts than the classical EWMA chart. Naveed et al. (2020, 2021) investigated the performance of the EEWMA chart using the multiple dependent state sampling and the repetitive sampling scheme, respectively. Finally, Nawaz et al. (2021) studied the robustness of the EWMA and DEWMA charts using the repetitive sampling scheme.

In this article, we propose and study a new exponentiated EWMA (Exp-EWMA) chart for normally distributed data to enhance the detection ability of the EWMA chart for small and moderate shifts. The main difference from the EWMA chart is that its smoothing parameter is not constant, but an exponentiated function.

The article is organized as follows. In Sect. 2, the existing CUSUM, EWMA and EEWMA charts are briefly reviewed, while in Sect. 3, the structure of the proposed Exp-EWMA chart is given. In Sect. 4, the run-length distribution of the Exp-EWMA chart is evaluated for both the zero-state and steady-state cases. In Sect. 5, we present the Exp-EWMA chart for the case where the process parameters are unknown. In Sect. 6, a comparison study with the CUSUM, EWMA and EEWMA charts is presented. In addition, we investigate the IC robustness of the proposed chart to non-normality and we compare it with those of its competitors in Sect. 7. Two illustrative examples are provided in Sect. 8 and conclusions are summarized in Sect. 8.

2 Existing control charts

In this section, we give a brief description of the structure of the CUSUM, EWMA and EEWMA control charts for monitoring shifts in the process mean.

2.1 The CUSUM control chart

Let $X_{tj}, t=1,2,3,\dots$ and $j=1,2,\dots ,n$, be the jth observation in the tth sample of size $n\ge 1$ and assume that $X_{tj}{\mathop {\sim }\limits ^{iid}} N(\mu ,\sigma ^2)$. The process is considered to be IC if $\mu =\mu _0$ and $\sigma =\sigma _0$. In this article, we focus on monitoring shifts in the process mean assuming that the process variance is IC and remains constant.

The upper and lower charting statistics of the CUSUM chart are given by

$$\begin{aligned} C_t^{+}=\text {max}\left[ 0,\left( \overline{X}_t - \mu _0\right) - k +C_{t-1}^{+}\right] \end{aligned}$$

and

$$\begin{aligned} C_t^{-}=\text {max}\left[ 0,- \left( \overline{X}_t - \mu _0\right) - k +C_{t-1}^{-}\right] , \end{aligned}$$

where $C_{0}^{+} = C_{0}^{-} =0$ and k is the reference value, computed by $k=\frac{\delta \sigma _0}{2\sqrt{n}}$, where $\delta$ is the shift (in terms of the process standard deviation $\sigma _0$) that we are interested to detect quickly. An OOC signal is given by the CUSUM chart if $C_{t}^{+} \ge H$ or $C_{t}^{-} \ge H$,

where$H=\frac{h\sigma _0}{\sqrt{n}}$ ($h>0$) is the decision interval.

2.2 The EWMA control chart

The charting statistic of the EWMA chart is denoted as

$$\begin{aligned} E_t=\lambda \overline{X}_t+(1-\lambda )E_{t-1}, \end{aligned}$$

where $E_0=\mu _0$ is the starting value and $0<\lambda \le 1$ is the smoothing parameter. The time-varying control limits and the centerline of the EWMA chart are given by

$$\begin{aligned} UCL_t / LCL_t = \mu _0 \pm L \frac{\sigma _0}{\sqrt{n}} \sqrt{\frac{\lambda }{2-\lambda }\left[ 1-\left( 1-\lambda \right) ^{2t}\right] }, CL=\mu _0, \end{aligned}$$

where $L>0$ is the width of the control limits. For large values of t, the asymptotic control limits are given by

$$\begin{aligned} UCL / LCL = \mu _0 \pm L \frac{\sigma _0}{\sqrt{n}} \sqrt{\frac{\lambda }{2-\lambda }}, CL=\mu _0, \end{aligned}$$

A process is considered to be OOC if any charting statistic falls on or outside the control limits. The EWMA chart reduces to the Shewhart chart for $\lambda =1$.

According to Steiner (1999), an EWMA chart is designed with time-varying control limits if there is an evidence that the shift occurs at the start (referred to as zero-state case). On the other hand, if the process is IC for several time periods and the shift occurs later (referred to as steady-state case), it is better to design a control chart with asymptotic control limits. The zero-state and steady-state performances of the EWMA chart are approximately similar for medium to large values of $\lambda$ (Lucas and Saccucci 1990). Small values of $\lambda$ are appropriate in detecting small shifts in the process mean while medium and large values are efficient in detecting moderate and large shifts, respectively.

2.3 The EEWMA control chart

The charting statistic of the EEWMA chart is denoted as

$$\begin{aligned} EE_t= \psi _{1} \overline{X}_t - \psi _{2} \overline{X}_{t-1} + (1-\psi _1+\psi _2) EE_{t-1}, \end{aligned}$$

where $0< \psi _{1} \le 1$, $0 \le \psi {2} < \psi _{1}$ and $\overline{X}_0=EE_{0}=\mu _0$. The time-varying control limits and the centerline of the EEWMA chart are given by

$$\begin{aligned}&UCL_t / LCL_t \\&= \mu _0 \pm L \frac{\sigma _0}{\sqrt{n}} \sqrt{\frac{\left( \psi _{1}^2+\psi _2^2\right) \left( 1-\alpha ^{2t} \right) }{2(\psi _{1}-\psi _{2})-(\psi _1-\psi _2)^2} - \frac{2 \alpha \psi _1 \psi _2 \left( 1-\alpha ^{2t-2} \right) }{2(\psi _{1}-\psi _{2})-(\psi _1-\psi _2)^2}},\\ \quad CL&=\mu _0, \end{aligned}$$

where $\alpha =1-\psi _1 + \psi _2$ and $L>0$ is the width of control limits. The asymptotic control limits of the EEWMA chart are given by

$$\begin{aligned} UCL / LCL = \mu _0 \pm L \frac{\sigma _0}{\sqrt{n}} \sqrt{\frac{\psi _{1}^2+\psi _2^2 -2 \alpha \psi _1 \psi _2 }{2(\psi _{1}-\psi _{2})-(\psi _1-\psi _2)^2}}, CL=\mu _0, \end{aligned}$$

The EEWMA chart reduces to the EWMA chart for $\psi _2=0$. A process is declared as OOC if any charting statistic falls on or outside the control limits. Otherwise, the process is said to be IC.

3 Structure of the proposed Exp-EWMA control chart

The proposed Exp-EWMA control chart is a modification of the EWMA chart where the smoothing parameter is not constant, but changes with t. More specifically, the charting statistic of the Exp-EWMA chart is defined as

$$\begin{aligned} Z_{t}=\lambda (t) \overline{X}_{t}+\left( 1-\lambda (t) \right) Z_{t-1}, \end{aligned}$$

(1)

where $\lambda (t)=e^{-f(t)}$ with $f(t) \ge 0$ continuous function of t, so that the sequence $\{f(t), t\ge 0\}$ converges and $Z_0=\mu _0$. In this article, we consider the following choice of f(t)

$$\begin{aligned} f(t)= & {} - ln \lambda + a^{\left( c+\frac{1}{t}\right) }, \end{aligned}$$

(2)

where $0< \lambda \le 1$ and $a,c \ge 0$ are two additional design parameters. Many functions of f(t) can be used. However, choosing the above function f(t), the smoothing parameter $\lambda (t)$ of the proposed chart changes over the time and lies between 0 and 1. Moreover, the function $\lambda (t)$ converges soon to the value $\lambda e^{-a^c}$.

Thus, the charting statistic of the Exp-EWMA chart can also be written as

$$\begin{aligned} Z_{t}=\lambda e^{- a^{\left( c+\frac{1}{t}\right) }} \overline{X}_{t}+\left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }} \right) Z_{t-1}. \end{aligned}$$

(3)

From (3), we have

$$\begin{aligned} Z_t= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t + \left( 1- \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) Z_{t-1}\\ {}= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t + \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \\{} & {} \left[ \lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\overline{X}_{t-1} + \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\right) Z_{t-2}\right] \\ {}= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t +\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \overline{X}_{t-1} + \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \times \\ {}{} & {} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\right) Z_{t-2} \\ {}= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t +\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \overline{X}_{t-1} + \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \times \\{} & {} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\right) \left[ \left( \lambda e^{- a^{\left( c+\frac{1}{t-2}\right) }}\right) \overline{X}_{t-2} + \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-2}\right) }}\right) Z_{t-3}\right] \\ {}= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t +\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \overline{X}_{t-1} + \lambda e^{- a^{\left( c+\frac{1}{t-2}\right) }}\\{} & {} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \times \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\right) \overline{X}_{t-2} +\left( 1-\lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) \\{} & {} \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-1}\right) }}\right) \left( 1-\lambda e^{- a^{\left( c+\frac{1}{t-2}\right) }}\right) Z_{t-3}. \end{aligned}$$

Proceeding in this way and since $Z_0=\mu _0$, we get

$$\begin{aligned} Z_t= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t+\sum _{i=1}^{t-1} \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) \overline{X}_i\\{} & {} + \prod _{j=1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \mu _0. \end{aligned}$$

The IC mean value and variance of the statistic $Z_t$ are respectively given by

$$\begin{aligned} E(Z_t|IC)= & {} \mu _0. \end{aligned}$$

(4)

and

$$\begin{aligned} Var(Z_t|IC)= & {} \left[ \left( \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) ^2+\sum _{i=1}^{t-1} \left\{ \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) \right\} ^2 \right] \frac{\sigma _0^2}{n}. \end{aligned}$$

(5)

The proof for the IC mean value is given in Appendix A. The time-varying control limits of the Exp-EWMA chart are denoted as

$$\begin{aligned} UCL_t / LCL_t = \mu _0 \pm L \sqrt{Var(Z_t|IC)}, CL=\mu _0, \end{aligned}$$

(6)

where $Var(Z_t|IC)$ is given by (5) and $L>0$ is the width of the control limits and it is chosen so that the IC performance of the control chart is equal to a pre-specified value.

The asymptotic variance of the charting statistic $Z_t$ is given by $\lim _{t \rightarrow +\infty } Var(Z_t|IC)=Var(Z_{\infty }|IC)$ which converges as $t \rightarrow +\infty$. For more details, see the Appendix B. Thus, the asymptotic control limits of the Exp-EWMA chart are denoted as

$$\begin{aligned} UCL / LCL = \mu _0 \pm L \sqrt{Var(Z_{\infty }|IC)}, CL=\mu _0, \end{aligned}$$

or equivalently

$$\begin{aligned} UCL / LCL = \mu _0 \pm H \frac{\sigma _0}{\sqrt{n}} , CL=\mu _0, \end{aligned}$$

(7)

where $H=L \frac{\sqrt{n}}{\sigma _0} \sqrt{Var(Z_{\infty }|IC)}$.

The Exp-EWMA chart is constructed by plotting the charting statistic $Z_t$ against the sample number t. A process is considered to be OOC if any charting statistic plots on or outside the control limits. We note that the proposed control chart reduces to the EWMA chart for $a=0$, while for $f(t) = \ln (t)$, we get the PM chart (see Appendix C). We also note that the proposed chart with $a=1$ reduces to the EWMA chart with smoothing parameter $\lambda /e$.

4 The run-length distribution of the Exp-EWMA control chart

The performance of a control chart is usually evaluated using the run-length distribution and its associated characteristics. The run-length is denoted as the number of statistics plotted on a chart before the chart triggers an OOC signal. The most common performance measure is the ARL. When the process is IC, the ARL is denoted as ARL₀ and it should be large to avoid any false alarm. On the other hand, when the process is OOC, the ARL is denoted as ARL₁ and it should be small to detect the shift quickly. Except of the ARL, other characteristics of the run-length distribution, such as the median, the standard deviation (regarded as MRL and SDRL, respectively) and several percentile points, are also evaluated to obtain more information about the run-length distribution. As we mentioned in the subsection 2.2, there are two types of ARL; the zero-state and the steady-state ARL. The first one is evaluated for the case where the shift occurs at the start, while the second one is evaluated for the case where the process is IC for several periods and the shift occurs later in the process.

The run-length characteristics of a control chart can be computed using the Markov chain approach, integral equations and Monte Carlo simulations. Due to the complexity of the charting statistic and the expression of time-varying control limits, in this article, we perform the latter method with 50.000 repetitions using the asymptotic control limits, given by (7). Without loss of generality, we consider individual measurements with $\mu _0=0$ and $\sigma _0=1$. Furthermore, the results are presented only for positive shifts as they are similar to those for negative shifts.

Fig. 1

Profiles $\lambda (t)$ versus t for a (a, c)=(0.5,0), b (0.5,0.5), c (0.75,0) and d (2,1)

The performance of the Exp-EWMA chart is investigated using different values of the design parameters ($a,c,\lambda$). More specifically, we set (a, c)=(0.5,0), (0.5,0.5), (0.75,0), (2,1) and $\lambda =0.05,0.10,0.25,0.50,0.75$. The profiles $\lambda (t)$ for $t=1,2,\dots ,50$ are presented in Fig. 1 from which we observe that $\lambda (t)$ is a decreasing (increasing) function for $a<1$ ($a>1$) and converges soon to the value $\lambda _{\infty }=\lambda e^{-a^c}$. Thus, it is expected that the steady-state performance of the Exp-EWMA chart to be equal to that of the EWMA chart with $\lambda _{\infty }$. This topic will be investigated later.

Table 1 Run-length characteristics for the Exp-EWMA control chart with $a=0.5$ and $c=0$ for the zero-state case

Table 2 Run-length characteristics for the Exp-EWMA control chart with $a=0.5$ and $c=0.5$ for the zero-state case

Table 3 Run-length characteristics for the Exp-EWMA control chart with $a=0.75$ and $c=0$ for the zero-state case

Table 4 Run-length characteristics for the Exp-EWMA control chart with $a=2$ and $c=1$ for the zero-state case

Tables 1-4 present the zero-state run-length characteristics of the Exp-EWMA chart given an ARL₀$\approx 500$ for the above values of design parameters ($a,c,\lambda$) and for shifts $\delta \in \{0.05, 0.10, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00, 3.00 \}$. More specifically, the first row of each cell presents the ARL and SDRL (given in parenthesis) while the second row shows the 5th, 25th, 50th (MRL), 75th and 95th percentile points of the run-length. From these tables, we observe that for specified values (a, c), the IC SDRL value (referred to as SDRL₀) and the 75th and 95th percentile points of the IC run-length distribution increase as the value of $\lambda$ also increases. On the other hand, the 5th, 25th and IC MRL (referred to as MRL₀) decrease as the value of $\lambda$ increases. It can be also seen that for (a, c)=(0.5,0), (0.5,0.5), (0.75,0) and for medium to large values of $\lambda$, the SDRL₀ value is larger than 500. This phenomenon may lead to a probability of false OOC signal (see Chan and Zhang 2000). Moreover, the run-length distribution is positively-skewed for the IC state and small to moderate shifts ($\delta \le 1$) as ARL>MRL, while it is less positively-skewed for larger shifts as in such cases ARL$\approx$MRL.

With regards to the OOC performance of the proposed chart for the zero-state case, we observe that a small value of $\lambda$ is preferred over larger values to detect small shifts of the process mean, while medium to large values of $\lambda$ are more effective for moderate to large shifts. For example, for (a, c)=(0.75,0), the Exp-EWMA chart with $\lambda =0.05$ outperforms the other charts for $\delta \le 0.25$ whereas the Exp-EWMA charts with $\lambda =0.10,0.25,0.50,0.75$ have very good detection ability for $\delta =0.50$, $\delta =0.75$, $1.00 \le \delta \le 1.25$ and $\delta \ge 1.50$, respectively. Finally, it should be noted that for specified values of $\lambda$ and $\delta$, the OOC performance of the proposed chart depends on the values of the design parameters (a, c). For instance, when $\lambda =0.05$ and $\delta =0.25$, the (ARL,MRL) values for (a, c)=(0.5,0), (0.5,0.5), (0.75,0) and (2,1) are (71.46,59), (73.25,59), (73.22,61) and (82.28,73).

Tables 5-8 present the steady-state run-length characteristics of the Exp-EWMA chart for the design parameters ($a,c,\lambda ,H$) and shifts used to the zero-state case. Comparing with the results of Tables 1-4, we observe that there are differences between the zero-state and steady-state run-length characteristics. More specifically, for $a<1$, the steady-state ARL₀ and the percentile points of the IC run-length distribution are smaller than the corresponding zero-state values for small values of $\lambda$ and vice versa for medium to large values. Furthermore, there is no a significant difference between the zero-state and steady-state SDRL₀ values. It should be also mentioned that for $a<1$ and small values of $\lambda$, the steady-state ARL₁ are larger than the corresponding zero-state ARL₁ for $\delta \ge 0.25$ or 0.50. For example, the zero-state ARL₁ values of the Exp-EWMA ($a=0.5,c=0,\lambda =0.10,H=0.3452$) chart are 400.33, 253.20, 75.46, 26.37, 14.89, 10.19, 7.67, 6.13, 4.35 and 2.77 for $\delta =0.05, 0.10, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00$ and 3.00, respectively (see Table 1), while the corresponding steady-state ARL₁ values are 389.37, 246.29, 76.01, 27.94, 16.65, 11.85, 9.22, 7.57, 5.63 and 3.80 (see Table 5). On the other hand, for $a<1$ and medium to large values of $\lambda$, the steady-state ARL₁ values are larger than the corresponding zero-state ARL₁ values over the entire range of shifts. For instance, the zero-state ARL₁ values of the Exp-EWMA ($a=0.75,c=0,\lambda =0.50,H=0.9395$) chart are 457.16, 362.65, 140.23, 38.03, 16.36, 9.53, 6.50, 4.90, 3.29 and 2.04 for $\delta =0.05, 0.10, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00$ and 3.00, respectively (see Table 3), while the corresponding steady-state ARL₁ values are 460.96, 367.75, 142.82, 39.30, 17.28, 10.26, 7.13, 5.47, 3.76 and 2.41 (see Table 7). With regards to the Exp-EWMA ($a=2,c=1$) chart, the steady-state ARL values are smaller than the corresponding zero-state ARL values due to the large difference between the zero-state and steady-state ARL₀ values.

Table 5 Run-length characteristics for the Exp-EWMA control chart with $a=0.5$ and $c=0$ for the steady-state case

Table 6 Run-length characteristics for the Exp-EWMA control chart with $a=0.5$ and $c=0.5$ for the steady-state case

Table 7 Run-length characteristics for the Exp-EWMA control chart with $a=0.75$ and $c=0$ for the steady-state case

Table 8 Run-length characteristics for the Exp-EWMA control chart with $a=2$ and $c=1$ for the steady-state case

5 Phase II Exp-EWMA chart

Up to this point, we have assumed that the quality characteristic of interest follows a normal distribution with known values of $\mu$ and $\sigma$. However, in many real-life applications, the IC values of $\mu$ and $\sigma$ are unknown and they are estimated from an IC Phase I sample before on-line monitoring starts in Phase II. We note that the Phase I sample is chosen when the process is thought to be IC. The estimates of $\mu _0$ and $\sigma _0$ are denoted by $\hat{\mu }_0$ and $\hat{\sigma }_0$, respectively and are used to estimate the starting value $Z_0$ and the control limits. Let $X_{tj}, t=1,2,\dots ,m$ and $j=1,2,\dots ,n$, be a sequence of IC Phase I observations which follow the $N(\mu _0,\sigma _0^2)$ distribution. The unbiased estimators for $\mu _0$ and $\sigma _0$ are given as

$$\begin{aligned} \hat{\mu }_0=\frac{\sum _{t=1}^{m} \sum _{j=1}^{n} X_{tj}}{m \cdot n} \end{aligned}$$

and

$$\begin{aligned} \hat{\sigma }_0=\frac{\sqrt{\frac{\sum _{t=1}^{m} \sum _{j=1}^{n} \left( X_{tj} - \overline{X}_t\right) ^2}{m \cdot (n-1)}}}{c_{4,m}}, \end{aligned}$$

where $c_{4,m}=\frac{\sqrt{2}\Gamma \left( \frac{m\left( n-1\right) +1}{2}\right) }{\sqrt{m(n-1)} \cdot \Gamma \left( \frac{m(n-1)}{2}\right) }$; see, for example Malela-Majika et al. (2022). The control limits of the Exp-EWMA chart are calculated using (7) where the estimates of $\mu _0$ and $\sigma _0$ are substituted for the unknown parameters.

The above methodology can used to implement the proposed chart in situations where the process parameters are unknown. Extensive simulations are required in order to investigate the effect of parameters estimation on the performance of the Exp-EWMA chart. This topic is out of the scope of this article and should be interest for future research.

6 Comparison study

In order to compare the OOC performance of different control charts, it is recommended to set a similar value of ARL₀, commonly 370 or 500, for all the charts. The chart with the smallest ARL₁ value in a specific shift is considered to be the most effective. Many authors, such as Chakraborti (2007), Teoh et al. (2017) and Qiao et al. (2022), recommended to compare different control charts using the OOC MRL (denoted as MRL₁) values by designing them under a similar value of MRL₀ because it provides a more meaningful interpretation than the ARL. Chan and Zhang (2000) suggested to take into account in the design of a chart not only the ARL₀, but also the SDRL₀ to avoid the large variation in the run-length distribution. In this article, the proposed Exp-EWMA chart is compared with the EWMA, CUSUM and EEWMA charts for both the zero-state and steady-state cases under similar IC run-length properties. For this reason, we chose the EWMA charts with $\lambda =0.05,0.10,0.25,0.50$ and the CUSUM charts with $k=0.125,0.5,1$ (CUSUM charts optimally designed to detect shifts of 0.25, 1.00 and 2.00, respectively) and subsequently, we found the appropriate EEWMA and Exp-EWMA charts with similar values of ARL₀, SDRL₀ and 5th, 25th, 50th, 75th, 95th percentile points (regarded as $P_5$, $P_{25}$, $P_{50}$, $P_{75}$ and $P_{95}$) of the IC run-length distribution. Moreover, the Exp-EWMA chart is compared with the EWMA ($\lambda _{\infty }$) chart in order to contrast their performances.

Table 9 presents the IC run-length characteristics of the competing charts for both the zero-state and steady-state cases. We note that first, we compare the EWMA chart and its modifications, ie the EEWMA and Exp-EWMA charts, and then, we compare the Exp-EWMA and CUSUM charts. From Table 9, we observe that choosing the appropriate control charts with similar zero-state IC run-length properties, the corresponding steady-state properties are similar.

Table 9 IC run-length characteristics of the competing charts for the zero-state and steady-state cases

Table 10 shows the ARL and MRL (in the parenthesis) values of the competing EWMA, Exp-EWMA and EEWMA charts for the zero-state case. There are four quadruple of competing charts where the first EWMA, the EEWMA and the Exp-EWMA charts have similar IC run-length characteristics, while the second EWMA chart is designed with $\lambda _{\infty }=\lambda e^{-a^c}$. For example, in the first quadruple of competing charts, the EWMA ($\lambda =0.05, L=2.613$), the EEWMA ($\psi _1=0.07, \psi _2=0.03, L=2.701$) and the Exp-EWMA ($\lambda =0.06, a=0.5, c=0.5, H=0.2986$) charts have similar IC run-length characteristics (see Table 9), while the other EWMA chart is designed with smoothing parameter equal to $\lambda _{\infty }$ and has different IC run-length characteristics from the other two charts.

Table 10 Zero-state ARL and MRL (in the parenthesis) comparison between EWMA, EEWMA and Exp-EWMA charts

Comparing the Exp-EWMA and EWMA charts with similar IC run-length characteristics, it is seen that the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.14$) charts are more effective than the EWMA ($\lambda =0.05$) and EWMA ($\lambda =0.10$) charts, respectively, for small shifts ($\delta \le 0.50$), while for the rest of the shifts, the competing charts perform similarly. Moreover, the Exp-EWMA ($a=0.75,c=0,\lambda =0.27$) chart has significantly smaller ARL₁ values than the EWMA ($\lambda =0.25$) chart for $\delta \le 1.00$ and smaller MRL₁ values for $\delta \le 0.75$. Finally, the Exp-EWMA ($a=0.75,c=0,\lambda =0.40$) chart is superior to the EWMA ($\lambda =0.50$) chart for $\delta \le 1.50$. To sum up, we conclude that the proposed Exp-EWMA chart is more effective than the EWMA chart for small shifts when both of them are designed to detect a small shift quickly, while its superiority increases as the value of shift, that we want the charts to detect optimally, increases.

Similar conclusions, as previously, are observed by comparing the Exp-EWMA and EEWMA charts. We conclude that the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.14$) charts are superior to the EEWMA ($\psi _1=0.07,\psi _2=0.03$) and EEWMA ($\psi _1=0.12, \psi _2=0.03$), respectively, for $\delta \le 0.50$, while for the rest range of shifts, the competing charts have similar OOC behaviour. Furthermore, the Exp-EWMA ($a=0.75,c=0,\lambda =0.27$) chart outperforms the EEWMA ($\psi _1=0.30, \psi _2=0.10$) chart for $\delta \le 1.25$, while the two charts perform similarly for the rest range of shifts. Finally, the ARL₁ and MRL₁ values of the Exp-EWMA ($a=0.75,c=0,\lambda =0.40$) chart are significantly smaller than those of the EEWMA ($\psi _1=0.60,\psi _2=0.20$) chart.

With regards to the comparison study between the Exp-EWMA and EWMA ($\lambda _{\infty }$) charts, we observe that the Exp-EWMA chart has smaller ARL₁ and MRL₁ values, especially for moderate to large shifts ($\delta \ge 0.50$), while it performs a slightly better for small shifts.

Table 11 presents the ARL and MRL values of the competing EWMA, Exp-EWMA and EEWMA charts for the steady-state case. The results of the comparison study between the Exp-EWMA and EWMA charts under similar IC run-length characteristics indicates that the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.14$) charts are more effective than the EWMA ($\lambda =0.05$) and EWMA ($\lambda =0.10$) charts for $\delta \le 0.25$ and $\delta \le 0.50$ respectively, and vice versa for the rest of the shifts. Furthermore, the Exp-EWMA ($a=0.75,c=0,\lambda =0.27$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.40$) charts have better detection ability than the EWMA ($\lambda =0.25$) and EWMA ($\lambda =0.50$) charts for $\delta \le 1.00$ and $\delta \le 1.50$ respectively, while the EWMA charts perform better for larger shifts.

With regards to the results about the comparison between Exp-EWMA and EWMA ($\lambda _{\infty }$) charts, it is seen that the two charts perform similarly for small to large shifts ($\delta \ge 0.25$). There are very small differences between the ARL₁ (and MRL₁) values due to the smaller ARL₀ (and MRL₀) of the EWMA charts.

Comparing the Exp-EWMA and EEWMA charts for the steady-state case, we observe that the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.14$) charts are more sensitive than the EEWMA ($\psi _1=0.07,\psi _2=0.03$) and EEWMA ($\psi _1=0.12,\psi _2=0.03$) charts for $\delta \le 0.25$ and $\delta \le 0.50$ respectively, and vice versa for the rest of the shifts. Moreover, the Exp-EWMA ($a=0.75,c=0,\lambda =0.27$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.40$) charts are more effective than the EEWMA ($\psi _1=0.30,\psi _2=0.10$) and EEWMA ($\psi _1=0.60,\psi _2=0.20$) charts for $\delta \le 1.00$ and $\delta \le 1.50$ respectively, while the EEWMA charts perform better for larger shifts.

As a general conclusion from the above comparison study, the proposed Exp-EWMA chart is more effective than the EWMA and EEWMA charts for small to moderate shifts for both zero-state and steady-state cases when the charts are designed under similar IC run-length characteristics. Furthermore, comparing the proposed chart with the EWMA ($\lambda _{\infty }$) chart, the first one is more sensitive for moderate to large shifts under the zero-state case, while the two charts have similar OOC performance under the steady-state case.

Table 11 Steady-state ARL and MRL (in the parenthesis) comparison between EWMA and Exp-EWMA charts

Table 12 shows the zero-state ARL and MRL values of the CUSUM and Exp-EWMA charts. It is seen that the Exp-EWMA ($a=0.75,c=0,\lambda =0.02$) chart is more sensitive than the CUSUM($k=0.125$) chart for $\delta \le 0.25$, while both of them have the same detection ability for moderate and large shifts. Similar to the previous comparison study, the superiority of the Exp-EWMA chart versus the CUSUM chart increases as the value of $\delta ^{*}$ increases. For example, the Exp-EWMA ($a=0.75,c=0,\lambda =0.21$) chart is more effective than the CUSUM($k=0.5$) chart for $\delta \le 1.00$, while for the rest of the shifts, the competing charts perform similarly. Moreover, the Exp-EWMA ($a=0.75,c=0,\lambda =0.25$) chart performs better than the CUSUM($k=1$) chart for $\delta \le 1.25$ and vice versa for $\delta \ge 2.00$.

Table 12 Zero-state ARL and MRL (in the parenthesis) comparison between CUSUM and Exp-EWMA charts

The steady-state ARL and MRL values of the competing CUSUM and Exp-EWMA charts are presented in Table 13. From this table, we observe that the Exp-EWMA ($a=0.75,c=0,\lambda =0.02$) chart performs better than the CUSUM($k=0.125$) chart only for very small shifts ($\delta \le 0.10$), while for the rest of the shifts, the CUSUM chart is more effective. Moreover, the Exp-EWMA ($a=0.75,c=0,\lambda =0.21$) chart is more sensitive than the CUSUM($k=0.5$) chart for $\delta \le 0.75$, while the latter chart has a slightly better OOC performance for the rest of the shifts. Finally, the Exp-EWMA ($a=0.75,c=0,\lambda =0.25$) chart outperforms the CUSUM($k=1$) chart for $\delta \le 1.25$ and vice versa for larger shifts.

Table 13 Steady-state ARL and MRL (in the parenthesis) comparison between CUSUM and Exp-EWMA charts

7 IC robustness

According to Human et al. (2011), the IC robustness of a control chart is a very important issue for its proper design implementation. A control chart is considered to be IC robust if its run-length distribution remains unchanged or nearly unchanged when the assumption of the underlying distribution, which is usually normality, is violated (Rocke 1989).

As we considered in the previous sections, the proposed chart has been designed assuming that the underlying distribution is the normal one. In order to study the IC robustness of the Exp-EWMA chart to non-normality and compare it with those of EWMA, EEWMA and CUSUM charts, we consider the following distributions: (i) the Student $t_{\nu }$ distribution with degrees of freedom $\nu =4$ and 8, (ii) the logistic $LG\left( 0,\sqrt{3}/\pi \right)$ distribution, (iii) the double exponential of Laplace $L\left( 0,1/\sqrt{2}\right)$ distribution and (iv) the gamma $G(\alpha ,\beta )$ distribution with $\alpha =1,2,4$ and $\beta =1$. We note that the Student and logistic distributions are symmetric with heavier tails than the normal distribution, while the Laplace distribution is symmetric with fatter tails than the normal. Moreover, the gamma distribution is positively skewed and was chosen to investigate the effect of skewness.

Table 14 presents the ARL₀ values of the control charts mentioned in the previous section for the above distributions. From this table, we conclude the following:

• 1. The proposed chart is more IC robust than the EWMA and EEWMA charts, while the EWMA chart has better IC robustness ability than the EEWMA chart.

• 2. The Exp-EWMA chart is more IC robust as the value of smoothing parameter $\lambda$ decreases. For example, the ARL₀ values of the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06$), Exp-EWMA ($a=0.75,c=0,\lambda =0.14$), Exp-EWMA ($a=0.75,c=0,\lambda =0.27$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.40$) charts under the $LG\left( 0,\sqrt{3}/\pi \right)$ distribution are 490.07, 472.41, 422.84 and 374.48, respectively. Similar findings can be seen for the EWMA and EEWMA charts.

• 3. In the case of $G(\alpha ,\beta )$ distribution, the Exp-EWMA chart is more IC robust as the value of parameter $\alpha$ increases. For example, the ARL₀ values of the Exp-EWMA ($a=0.75,c=0,\lambda =0.14$) chart under the G(1, 1), G(2, 1) and G(4, 1) distributions are 460.31, 481.99 and 490.83, respectively. This happens because the gamma distribution approaches the normal one for large values of $\alpha$.

• 4. Comparing the IC robustness of the Exp-EWMA and CUSUM charts, we observe that the Exp-EWMA ($a=0.75,c=0,\lambda =0.02$) and CUSUM($k=0.125$) charts have similar IC robustness abilities, while the Exp-EWMA ($a=0.75,c=0,\lambda =0.21$) and Exp-EWMA ($a=0.75,c=0,\lambda =0.25$) charts are more IC robust than the CUSUM($k=0.5$) and CUSUM($k=1$) charts, respectively.

Table 14 IC robustness of different control charts

Table 15 Simulated dataset and charting statistics

8 Illustrative Examples

In this section, we present two examples, one with simulated dataset and one with real dataset, in order to demonstrate the application of the proposed chart.

Fig. 2

Control charts for simulated dataset a EWMA b CUSUM c Exp-EWMA d EEWMA

8.1 Simulated dataset

In this example, we consider 10 individual measurements from a N(0, 1) distribution and 40 from a N(0.25, 1) distribution. The IC values of the process mean and standard deviation are $\mu _0=0$ and $\sigma _0=1$. Thus, the process is IC for the first 10 observations and a shift of $\delta =0.25$ in the process mean is added to the last 40 observations (steady-state case). Table 15 presents the dataset as $X_t$. Setting an ARL₀ $\approx 500$, we construct the EWMA ($\lambda =0.05,L=2.613$), the CUSUM($k=0.125,H=13.135$), the Exp-EWMA ($a=0.5,c=0.5,\lambda =0.06,H=0.2986$) and EEWMA ($\psi _1=0.07,\psi _2=0.03,L=2.701$) charts. The EWMA, Exp-EWMA and EEWMA charts have similar IC run-length characteristics (see Table 9), while the specific CUSUM chart is optimally designed to detect a shift of $\delta =0.25$. The charting statistics $E_t$, $C_{t}^{+}$ (only the upper statistic), $Z_t$ and $EE_t$ of the EWMA, CUSUM, Exp-EWMA and EEWMA charts are also provided in Table 15, where those that lie above the UCL are in bold print. The control charts are displayed in Fig. 2. The Exp-EWMA chart gives OOC signals at the 48th and 49th charting statistics while the other charts cannot detect the shift.

8.2 Real dataset

In this example, we use a dataset, given by Montgomery (2013), about the inside diameter measurements (in mm) from automobile engine piston rings. The dataset is provided in Table 16 and consists of 40 samples each of size $n=5$. The first 25 samples represent the Phase I process and are used to compute the IC process mean and standard deviation; those are $\mu _0=74.001$mm and $\sigma _0=0.009424$mm. The second 15 samples represent the Phase II process. Setting an ARL₀ $\approx 500$, we construct the EWMA ($\lambda =0.50,L=3.071$), the CUSUM($k=1,h=2.665$), the Exp-EWMA ($a=0.75,c=0,\lambda =0.40,H=0.8195$) and the EEWMA ($\psi _1=0.60,\psi _2=0.20,L=3.085$) charts. The EWMA, Exp-EWMA and EEWMA charts have similar IC run-length characteristics, while the CUSUM chart is optimally designed to detect a shift of $\delta =2.00$. The charting statistics of the control charts are also presented in Table 16, where the bold prints indicate those that lie above the UCL. The control charts are displayed in Fig. 3. We observe that all charts detect the shift after 35 samples. Thus, in this example, the proposed chart performs similarly to its competitors.

Table 16 Piston ring diameter data and values of charting statistics

Fig. 3

Control charts for for the piston rings dataset a EWMA b CUSUM c Exp-EWMA d EEWMA

9 Conclusion

In this article, we extended the EWMA chart and proposed the exponentiated EWMA (Exp-EWMA) chart where its smoothing parameter is not constant, but changes with the sample number. Although the construction of the Exp-EWMA chart is more complicated than that of the traditional EWMA chart, the proposed chart is found more effective especially for more shifts and more IC robust for non-normal distributions.

The existing EWMA and PM charts are special cases of the new chart. Performing the Monte Carlo simulation approach, we computed the zero-state and steady-state run-length characteristics of the Exp-EWMA chart using the asymptotic control limits. A comparison study with the EWMA, EEWMA and CUSUM charts under similar IC run-length characteristics indicated that the proposed Exp-EWMA chart has better detection ability for small shifts when the competing charts are designed to detect a small shift quickly. The superiority of the Exp-EWMA chart versus its competitors increases as the amount of shift, that we want the charts to detect quickly, increases. Moreover, the Exp-EWMA chart is superior to the EWMA ($\lambda _{\infty }$) chart for the zero-state case, while the two charts perform similarly for the steady-state case. Finally, we investigated the IC robustness of the proposed chart for several non-normal distributions and we showed that the Exp-EWMA chart has better IC robustness ability than its competitors.

The current paper may be extended to design the Exp-EWMA chart for monitoring changes in the process variance, as well as changes in both location and scale parameters. Moreover, the effect of parameters estimation on the performance of the Exp-EWMA chart is proposed as a future topic. Another interest topic of future research should be the investigation of optimally designed Exp-EWMA charts for specific shifts as there are three design parameters ($\lambda ,a,c$).

Appendices

Appendix A

Proof of $E(Z_t|IC)= \mu _0$.

The charting statistic of the proposed Exp-EWMA chart is given by

$$\begin{aligned} Z_t= & {} \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\overline{X}_t+\sum _{i=1}^{t-1} \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) \overline{X}_i\\{} & {} + \prod _{j=1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \mu _0 \end{aligned}$$

Moreover, it is to noted that $E(\overline{X}_t|IC)=\mu _0$. Therefore,

$$\begin{aligned}{} & {} E(Z_t|IC)\\{} & {} \quad = \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}E(\overline{X}_t)+\sum _{i=1}^{t-1} \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) E(\overline{X}_i)\\{} & {} \qquad \qquad + \prod _{j=1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \mu _0\\{} & {} \quad = \left[ \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}+\sum _{i=1}^{t-1} \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) \right. \\{} & {} \qquad \qquad \left. + \prod _{j=1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \right] \mu _0 \\{} & {} \quad = \left[ \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}+\sum _{i=1}^{t-1} \left[ \left( \prod _{j=i+1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \right) - \left( \prod _{j=i}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \right) \right] \right. \\{} & {} \quad \quad \left. +\prod _{j=1}^{t}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j}\right) }}\right) \right] \mu _0 = \mu _0. \end{aligned}$$

Hence proved.

Appendix B

Convergence of $\lim _{t \rightarrow +\infty } Var(Z_t|IC)$.

The IC variance of the statistic $Z_t$ is given by

$$\begin{aligned} Var(Z_t|IC)= & {} \left[ \left( \lambda e^{- a^{\left( c+\frac{1}{t}\right) }}\right) ^2+\sum _{i=1}^{t-1} \left\{ \lambda e^{- a^{\left( c+\frac{1}{i}\right) }}\left( \prod _{j=i}^{t-1}\left( 1-\lambda e^{- a^{\left( c+\frac{1}{j+1}\right) }}\right) \right) \right\} ^2 \right] \frac{\sigma _0^2}{n}, \end{aligned}$$

where $0<\lambda \le 1$ and $a,c\ge 0$ We note that

$$\begin{aligned} \begin{array}{cc} a^c \le a^{(c+\frac{1}{j})}\le a^{c+1},&{}\text{ if } a > 1,\\ a^{c+1} \le a^{(c+\frac{1}{j})}\le a^{c},&{}\text{ if } 0< a < 1. \end{array} \end{aligned}$$

This implies

$$\begin{aligned} \begin{array}{cc} m_1 \le e^{-a^{(c+\frac{1}{j})}}\le M_1,&{}\text{ if } a > 1,\\ M_1 \le e^{-a^{(c+\frac{1}{j})}}\le m_1,&{}\text{ if } 0< a < 1, \end{array} \end{aligned}$$

where $m_1=e^{-a^c}$ and $M_1=e^{-a^{c+1}}$.

For $a > 1$, we have

$$\begin{aligned} \frac{n}{\sigma _0^2}Var(Z_t|IC)\le & {} \lambda ^2M_1^2 + \sum _{i=1}^{t-1} \lambda ^2 M_1^2 (1-\lambda m_1)^{2(t-i)} = \lambda ^2M_1^2 \sum _{i=0}^{t-1} (1-\lambda m_1)^{2i}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{n}{\sigma _0^2} Var(Z_{\infty }|IC)\le & {} \lambda ^2M_1^2 \sum _{i=0}^{\infty } (1-\lambda m_1)^{2i} = \frac{\lambda ^2M_1^2}{1-(1-\lambda m_1)^2}. \end{aligned}$$

Moreover,

$$\begin{aligned} \frac{n}{\sigma _0^2} Var(Z_t|IC)\ge & {} \lambda ^2m_1^2 + \sum _{i=1}^{t-1} \lambda ^2 m_1^2 (1-\lambda M_1)^{2(t-i)} = \lambda ^2m_1^2 \sum _{i=0}^{t-1} (1-\lambda M_1)^{2i}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{n}{\sigma _0^2} Var(Z_{\infty }|IC)\ge & {} \lambda ^2m_1^2 \sum _{i=0}^{\infty } (1-\lambda M_1)^{2i} = \frac{\lambda ^2m_1^2}{1-(1-\lambda M_1)^2}. \end{aligned}$$

Combining the above, we get

$$\begin{aligned} \frac{\lambda ^2m_1^2}{1-(1-\lambda M_1)^2}\le & {} \frac{n}{\sigma _0^2} Var(Z_{\infty }|IC)\; \le \;\frac{\lambda ^2M_1^2}{1-(1-\lambda m_1)^2}. \end{aligned}$$

Similarly, for $0<a < 1$, we get

$$\begin{aligned} \frac{\lambda ^2M_1^2}{1-(1-\lambda m_1)^2}\le & {} \frac{n}{\sigma _0^2} Var(Z_{\infty }|IC)\; \le \; \frac{\lambda ^2m_1^2}{1-(1-\lambda M_1)^2}. \end{aligned}$$

Appendix C

Progressive mean as a special case of the Exp-EWMA chart

The charting statistic of the PM chart is denoted as

$$\begin{aligned} PM_t=\frac{\sum _{k=1}^{t} \overline{X}_k}{t}, \end{aligned}$$

and the typical L-sigma control limits are denoted as

$$\begin{aligned} UCL_t / LCL_t = \mu _0 \pm L \frac{\sigma _0}{\sqrt{t}} , CL=\mu _0, \end{aligned}$$

Abbas et al. (2013) suggested the use of an arbitrary function $g(t)=t^{0.2}$ in the control limits, so that they are narrower for large values of t.

It is to be noted that for $f(t)=\ln (t)$, we have

$$\begin{aligned} e^{-f(t)} = \frac{1}{t} \end{aligned}$$

and

$$\begin{aligned} e^{-f(i)}\prod _{j=i}^{t-1}\left( 1- e^{- f(j+1)}\right) =\frac{1}{i} \prod _{j=i}^{t-1}\left( 1- \frac{1}{j+1}\right) = \frac{1}{t}. \end{aligned}$$

Therefore, we can write

$$\begin{aligned} Z_t=\frac{\sum _{k=1}^t \overline{X}_k}{t} \end{aligned}$$

Thus, for $f(t)=\ln (t)$, the Exp-EWMA chart reduces to the PM chart.