A Nonparametric Double Homogeneously Weighted Moving Average Signed-Rank Control Chart for Monitoring Location Parameter

Abstract

Nonparametric control charts are widely used in many manufacturing processes when there is a lack of knowledge about the distribution that the quality characteristic of interest follows. If there is evidence that the unknown distribution is symmetric, then the signed-rank statistic is preferred over other nonparametric statistics because it makes control charts more efficient. In this article, a nonparametric double homogeneously weighted moving average control chart based on the signed-rank statistic, namely, the DHWMA-SR chart, is introduced for monitoring the location parameter of an unknown, continuous and symmetric distribution. Monte Carlo simulations are used to study the run-length distribution of the proposed chart. A performance comparison study with the EWMA-SR, DEWMA-SR and HWMA-SR charts indicates that the DHWMA-SR chart is more effective under the zero-state scenario, while its steady-state performance is poor. Finally, two illustrative examples are given to demonstrate the application of the proposed chart.

1. Introduction

Statistical Process Control (SPC) is implemented in many manufacturing and non-manufacturing processes and aims to identify the assignable causes of unnatural variability. Control charts are the most important tool of SPC and are used for on-line process monitoring. There are two types of control charts: the memory-less charts, such as the Shewhart-type charts, introduced by Shewhart [1], and the memory-type charts, such as the cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) charts proposed by Page [2] and Roberts [3], respectively. The Shewhart-type control charts are easy to implement and effective for detecting large shifts in the process parameters but ineffective for small and moderate shifts. On the other hand, memory-type control charts are more complex to design but efficient for small and moderate shifts.

The design of control charts is usually based on the assumption that the quality characteristic of interest is a random variable that follows a known distribution, commonly, the normal distribution. However, in many cases, this assumption is violated, since little information about the underlying distribution is known. In such cases, the implementation of parametric control charts may lead to unreliable results about the process. For this reason, nonparametric (or distribution-free) control charts have been developed by many authors. There are two types of nonparametric statistics that are used for monitoring shifts in the location parameter (median) when the target value is known a priori: the sign statistic and the signed-rank statistic. The sign statistic requires only the assumption of continuity for the underlying distribution, while the signed-rank statistic requires the additional assumption of symmetry. As a result, nonparametric control charts based on the signed-rank statistic are usually more efficient than those based on the sign statistic [4].

Several nonparametric control charts based on the signed-rank statistic have been developed by many authors. An overview of them up to 2018 is presented by Chakraborti and Graham [5]. However, many researchers continue to study and propose new schemes. Raza et al. [6] proposed the nonparametric double EWMA chart based on the signed-rank statistic (DEWMA-SR), which is a mixture of two EWMA-SR charts, and they found it more effective than the EWMA-SR and several nonparametric sign charts for small shifts. Alevizakos et al. [7] presented the double generally weighted moving average chart based on the signed-rank statistic (DGWMA-SR) as an effort to improve the performance of the GWMA-SR chart for small shifts. Raza et al. [8] developed nonparametric homogeneously weighted moving average schemes based on the sign (HWMA-SN) and signed-rank (HWMA-SR) statistics. Comparing to the nonparametric EWMA and CUSUM schemes, the nonparametric HWMA schemes were found to be more effective for small and (in some cases) moderate shifts. Perdikis et al. [9] presented the EWMA-SR chart for count data (CEWMA-SR) where the charting statistics are positive integers. The CEWMA-SR chart was found to be more effective than the CUSUM-SR, EWMA-SR and GWMA-SR charts for moderate to large shifts. Perdikis et al. [10] developed a modified nonparametric EWMA-SR chart, namely, the C-WRS EWMA chart, and presented a technique that guarantees steady run-length properties. Alevizakos et al. [11] introduced the nonparametric triple EWMA chart based on the signed-rank statistic (TEWMA-SR) to enhance the detection ability of the EWMA-SR chart. It was shown that the TEWMA-SR chart outperforms the CUSUM-SR, EWMA-SR and DEWMA-SR charts for small shifts. Tang et al. [12] presented a discrete state nonparametric adaptive EWMA chart based on the signed-rank statistic (ADSEWMA-SR). The above works are referred to nonparametric control charts based on the signed-rank statistic. Some recently published articles about nonparametric control charts based on the sign statistic are those of Castagliola et al. [13], Tang et al. [14] and Alevizakos et al. [15]. An overview of nonparametric control charts is provided by Qiu [16] (chapters 8 and 9), Chakraborti and Graham [4,5], Koutras and Triantafyllou [17] and Triantafyllou and Ram [18], while Qiu [19] gives some interesting perspectives on issues related to the nonparametric SPC.

Abid et al. [20] and Riaz et al. [21] proposed the double and the triple HWMA charts, respectively. However, as Alevizakos et al. [22] and Knoth et al. [23] mentioned, these schemes are reparameterizations of the HWMA chart. Similarly, the nonparametric double HWMA chart based on the sign statistic and the triple HWMA chart under ranked set sampling, proposed by Riaz et al. [24] and Zhang et al. [25], respectively, are reparameterizations of the HWMA schemes. Alevizakos et al. [22] developed a double HWMA (DHWMA) chart for normally distributed data to enhance the performance of the HWMA chart [26] for small and moderate shifts based on the DEWMA technique. It should be noted that Adeoti and Koleoso [27] proposed an hybrid HWMA (HHWMA) chart, but unfortunately, the mathematical expression of the variance of the charting statistic was incorrect. Consequently, Malela-Majika et al. [28] provided the correct version of the HHWMA chart. A literature review of the HWMA scheme and its modifications is presented by Letshedi et al. [29]. Alevizakos et al. [30] presented the nonparametric DHWMA sign chart where it was found more effective than other nonparametric sign control charts, especially for small shifts in the location parameter. Motivated by [22,30], in this article, we propose a nonparametric DHWMA control chart based on the signed-rank statistic (regarded as DHWMA-SR chart) for monitoring shifts in the location parameter of a continuous and symmetric distribution.

The article is organized as follows. In Section 2, we present the structure of nonparametric control charts based on the signed-rank statistic, while Section 2 presents the proposed DHWMA-SR chart. The in-control (IC) and out-of-control (OOC) performances of the DHWMA-SR chart for the zero-state and steady-state scenarios are provided in Section 4 and Section 5, respectively. A comparison study of the proposed chart against other nonparametric charts and the parametric DHWMA-$\overline{X}$ chart is given in Section 6. Two examples are discussed in Section 7 to demonstrate the implementation of the DHWMA-SR chart. Finally, conclusions and recommendations are provided in Section 8.

2. Design Structure of Nonparametric Signed-Rank Control Charts

2.1. The Signed-Rank Statistic

Let X be the quality characteristic of interest following an unknown, continuous and symmetric distribution around the median $\theta$. Moreover, suppose that ${\theta }_0$ denotes the known value of $\theta$ and the process is considered to be IC if $\theta ={\theta }_0$. It should be noted that this value is equal to the IC mean value ${\mu }_0$ because the distribution is symmetric. Assume that a subgroup $\{X_{1t},X_{2t},\dots ,X_{nt}\}$ of size $n>1$ is collected at at time $t=1,2,\dots$ Let $R_{it}^{+}$, $i=1,2,\dots ,n$, denote the rank of the absolute value of the differences $|X_{it}-{\theta }_0|$, $i=1,2,\dots ,n$, within the t-th sample. The signed-rank (SR) statistic is defined as

$$S{R}_t=\sum _{i=1}^n\mathrm{sign}\left(X_{it}-{\theta }_0\right)R_{it}^{+},t=1,2,\dots ,$$

(1)

where $\mathrm{sign}\left(x\right)=1,0,-1$ if $x>0,x=0$ or $x<0$, respectively. The $S{R}_t$ statistic is the difference between the sum of the ranks of the absolute differences corresponding to the positive and the negative differences, i.e., $S{R}_t=T_t^{+}-T_t^{-}$. Furthermore, the $S{R}_t$ statistic can also be expressed as $S{R}_t=2T_t^{+}-n(n+1)/2$. The IC values of mean and variance of the $S{R}_t$ statistic are zero and $n(n+1)(2n+1)/6$, respectively. More information about the distribution of statistics $T_t^{+}$ and $S{R}_t$ can be found in [4,5,31].

2.2. The EWMA-SR Control Chart

The charting statistic of the EWMA-SR chart is defined by

$$E_t=\lambda S{R}_t+(1-\lambda )E_{t-1},$$

where $E_0=E\left(S{R}_t\right|IC)=0$ and $0<\lambda \le 1$ is the smoothing parameter. The upper and lower time-varying control limits of the EWMA-SR chart are given by

$$UC{L}_t/LC{L}_t=\pm L_E\sqrt{{\displaystyle \frac{n(n+1)(2n+1)}{6}}{\displaystyle \frac{\lambda }{2-\lambda }}[1-{(1-\lambda )}^{2t}]},$$

where $L_E>0$ is the coefficient of the control limits, which has to be computed so that the attained IC average run-length (ARL₀) is close to a desired value.

The EWMA-SR chart reduces to the Shewhart-SR chart when $\lambda =1$. A process is considered to be OOC if $E_t$ plots beyond the control limits. Otherwise, the process is considered to be IC and no shift has been occurred.

2.3. The DEWMA-SR Control Chart

The DEWMA-SR chart is a mixture of two EWMA-SR charts and its charting statistic is defined via the system of equations

$$\left\{\begin{array}{c}{E}_t={\lambda }_{1}S{R}_t+(1-{\lambda }_1)E_{t-1},\hfill \\ D_t={\lambda }_2{E}_t+(1-{\lambda }_2)D_{t-1},\hfill \end{array}\right.$$

where $E_0=D_0=E\left(S{R}_t\right|IC)=0$ and $0<{\lambda }_1,{\lambda }_2\le 1$ are the smoothing parameters. For the case where ${\lambda }_1={\lambda }_2=\lambda$, the IC expected value and variance of the statistic $D_t$ are given by

$$E\left(D_t|IC\right)=0$$

and

$$Var\left(D_t|IC\right)={\displaystyle \frac{{\lambda }^4[1+k-{(t+1)}^2{k}^t+(2t^2+2t-1)k^{t+1}-t^2{k}^{t+2}]}{{[1-k]}^3}}{\displaystyle \frac{n(n+1)(2n+1)}{6}},$$

where $k={(1-\lambda )}^2$. The time-varying control limits of the DEWMA-SR chart are given by

$$UC{L}_t/LC{L}_t=\pm L_D\sqrt{Var\left(D_t\right|IC)},$$

where $L_D>0$ is the coefficient of the control limits and, similar, to the EWMA-SR chart, it has to be computed so that the attained ARL₀ is close to a desired value.

The DEWMA-SR chart reduces to the EWMA-SR chart when ${\lambda }_1=1$ or ${\lambda }_2=1$ and to the Shewhart-SR chart when ${\lambda }_1={\lambda }_2=1$. A process is considered to be OOC if $D_t\le LC{L}_t$ or $D_t\ge UC{L}_t$. Otherwise, the process is said to be IC.

2.4. The HWMA-SR Control Chart

The charting statistic of the HWMA-SR chart gives a specific weight to the most current sample and the remaining is equally distributed among the past samples. More specifically, the charting statistic of the HWMA-SR chart is defined as

$$\begin{array}{c}\hfill H_t=\lambda S{R}_t+(1-\lambda ){\overline{SR}}_{t-1},\end{array}$$

where ${\overline{SR}}_0=0$, ${\overline{SR}}_{t-1}={\displaystyle \frac{{\sum }_{k=1}^{t-1}S{R}_k}{t-1}}$ and $0<\lambda \le 1$ is the smoothing parameter.

The control limits of the HWMA-SR chart are given by

$$\begin{array}{ccc}\hfill UC{L}_t/LC{L}_t& =& \left\{\begin{array}{cc}\pm L_H\lambda \sqrt{{\displaystyle \frac{n(n+1)(2n+1)}{6}}}\hfill & ,\mathrm{for}t=1,\hfill \\ \pm L_H\sqrt{\left({\lambda }^2+{\displaystyle \frac{{\left(1-\lambda \right)}^2}{t-1}}\right){\displaystyle \frac{n(n+1)(2n+1)}{6}}}\hfill & ,\mathrm{for}t>1,\hfill \end{array}\right.\hfill \end{array}$$

where $L_H>0$ is the coefficient of the control limits. The HWMA-SR chart reduces to the Shewhart-SR chart when $\lambda =1$. The process is considered to be OOC if any $H_t$ plots on or outside the control limits; otherwise, the process is considered to be IC.

3. The DHWMA-SR Control Chart

The DHWMA-SR chart is a mixture of two HWMA-SR charts and its charting statistic $D{H}_t$ is defined via the systems of equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{H}_t=\lambda S{R}_t+(1-\lambda ){\overline{SR}}_{t-1},\hfill \\ D{H}_t=\lambda H_t+(1-\lambda ){\overline{H}}_{t-1},\hfill \end{array}\right.\end{array}$$

(2)

where $0<\lambda \le 1$ is the smoothing parameter, ${\overline{SR}}_{t-1}={\displaystyle \frac{{\sum }_{k=1}^{t-1}S{R}_k}{t-1}}$, ${\overline{H}}_{t-1}={\displaystyle \frac{{\sum }_{k=1}^{t-1}{H}_k}{t-1}}$ and ${\overline{SR}}_0={\overline{H}}_0=E\left(S{R}_t|IC\right)=0$.

For $t=1$, we have

$$\begin{array}{c}\hfill D{H}_1={\lambda }^{2}S{R}_1,\end{array}$$

while for $t=2$, it is

$$\begin{array}{c}\hfill D{H}_2={\lambda }^{2}S{R}_2+2\lambda (1-\lambda )S{R}_1.\end{array}$$

For $t>2$, the charting statistic $D{H}_t$ simplifies to

$$\begin{array}{c}\hfill D{H}_t={\lambda }^{2}S{R}_t+{\displaystyle \frac{2\lambda (1-\lambda )}{t-1}}S{R}_{t-1}+{\displaystyle \frac{(1-\lambda )}{t-1}}\sum _{u=1}^{t-2}\left(2\lambda +(1-\lambda )\sum _{k=u}^{t-2}{\displaystyle \frac{1}{k}}\right)S{R}_u.\end{array}$$

The IC expected value of the charting statistic $D{H}_t$ is equal to 0 and its IC variance is given by

$$\begin{array}{ccc}\hfill Var\left(D{H}_t\right)& =& \left\{\begin{array}{cc}{\displaystyle \frac{{\lambda }^{4}n(n+1)(2n+1)}{6}}\hfill & ,\mathrm{for}t=1,\hfill \\ {\lambda }^2\left({\lambda }^2+4{(1-\lambda )}^2\right){\displaystyle \frac{n(n+1)(2n+1)}{6}}\hfill & ,\mathrm{for}t=2,\hfill \\ \begin{array}{cc}& \left[{\lambda }^4+{\displaystyle \frac{4{\lambda }^2{(1-\lambda )}^2}{{(t-1)}^2}}+\right.\hfill \\ & \left.{\displaystyle \frac{{(1-\lambda )}^2}{{(t-1)}^2}}\sum _{u=1}^{t-2}{\left(2\lambda +(1-\lambda )\sum _{k=u}^{t-2}{\displaystyle \frac{1}{k}}\right)}^2\right]{\displaystyle \frac{n(n+1)(2n+1)}{6}}\hfill \end{array}\hfill & ,\mathrm{for}t>2.\hfill \end{array}\right.\hfill \end{array}$$

(3)

For more details about the algebraic derivation for the mean and variance of DHWMA statistic, readers can consult [22]. The control limits of the DHWMA-SR chart are given by

$$\begin{array}{c}\hfill UC{L}_t/LC{L}_t=\pm L_{DH}\sqrt{Var\left(D{H}_t\right)},\end{array}$$

(4)

where $L_{DH}>0$ is the coefficient of the control limits which has to be calculated so that the attained ARL₀ is close to a desired ARL₀ and $Var\left(D{H}_t\right)$ is given by Equation (3). The DHWMA-SR chart is constructed by plotting the statistic $D{H}_t$ versus t. If any of $D{H}_t$ plots on or outside the control limits, a signal is given and the process is considered to be OOC. Otherwise, if $LC{L}_t<D{H}_t<UC{L}_t$, the process is said to be IC and the charting procedure continuous. Note that the proposed chart reduces to the Shewhart-SR chart when $\lambda =1$.

4. Performance Analysis for the Zero-State Scenario

4.1. Design and Implementation

The run-length distribution and its associated characteristics are usually used in order to measure the performance of a control chart. The run-length is a discrete random variable and is defined as the number of charting statistics that must be plotted on a control chart in order for the chart to give an OOC signal. The most popular metric of a control chart’s performance is the ARL which is the expected number of charting statistics that must be plotted on a chart before the first OOC signal [32]. For an efficient control chart, the ARL₀ should be large to avoid any false alarm, while the OOC ARL (ARL₁) should be small to detect the shift quickly. Other characteristics of the run-length distribution, such as the standard deviation (SDRL) and the median (MDRL) of the run-length, are recommended to use to obtain more information about the run-length distribution.

The run-length characteristics of a control chart can be evaluated via Markov chains, integral equations and Monte Carlo simulations. In this study, we perform the latter method because the first two methods cannot be easily applied to the proposed scheme due to the complexity of the charting statistic $D{H}_t$ and the form of control limits [22,30]. The simulation algorithm includes the following steps:

Step 1: 

For a specified value of $n>1$ and a shift $\delta$, generate 10,000 random variables from any continuous and symmetric distribution.

Step 2: 

For a desired value of ARL₀, specify the design parameters $\lambda$ and $L_{DH}$.

Step 3: 

Compute the $S{R}_t$ and $D{H}_t$ statistics using Equations (1) and (2), respectively.

Step 4: 

Compute the control limits of the DHWMA-SR chart using Equation (4).

Step 5: 

Compare each charting statistic with the corresponding control limits. The DHWMA-SR chart gives a signal if any $D{H}_t$ plots beyond the control limits.

Step 6: 

Repeat Steps 1 to 5 for 50,000 times and compute the run-length characteristics from the above values.

In order to evaluate the $L_{DH}$ values of the proposed chart, so that the ARL₀ be equal to a desired one, we perform the above algorithm using $\delta =0$.

Table 1. Values of $L_{DH}$ for the DHWMA-SR chart for different (n, $\lambda$) combinations to achieve a desired ARL₀.

		ARL0						ARL0
$\mathit{n}$	$\mathit{\lambda }$	200	300	370	500	$\mathit{n}$	$\mathit{\lambda }$	200	300	370	500
5	0.10	0.897	0.955	1.000	1.067	10	0.10	0.931	1.023	1.071	1.153
	0.14	1.135	1.221	1.280	1.368		0.14	1.195	1.324	1.395	1.506
	0.15	1.204	1.293	1.356	1.448		0.15	1.269	1.403	1.479	1.598
	0.17	1.310	1.437	1.492	1.583		0.17	1.404	1.567	1.650	1.782
	0.20	1.491	1.622	1.688	1.766		0.20	1.617	1.795	1.890	2.033
	0.25	1.756	1.867	1.915	1.971		0.25	1.944	2.134	2.229	2.363
	0.30	1.939	2.007	2.027	2.052		0.30	2.199	2.369	2.451	2.555
15	0.10	0.941	1.035	1.085	1.173	20	0.10	0.948	1.043	1.094	1.178
	0.14	1.207	1.342	1.420	1.534		0.14	1.216	1.352	1.432	1.553
	0.15	1.279	1.422	1.504	1.631		0.15	1.285	1.435	1.521	1.650
	0.17	1.419	1.590	1.679	1.820		0.17	1.431	1.603	1.700	1.843
	0.20	1.641	1.831	1.936	2.090		0.20	2.652	1.850	1.958	2.118
	0.25	1.980	2.185	2.296	2.442		0.25	1.998	2.212	2.325	2.481
	0.30	2.243	2.438	2.531	2.654		0.30	2.269	2.470	2.570	2.699

presents the $L_{DH}$ values of the DHWMA-SR chart for $n=5,10,15,20$ and $\lambda =0.10,0.14,0.15,0.17,0.20,0.25,0.30$, so that the desired ARL₀ be equal to 200, 300, 370 and 500. These values were computed using the standard normal distribution; however, any other continuous and symmetric distribution can be used. From Table 1, we observe that for a specified value of n ($\lambda$), the value of $L_{DH}$ increases as the value of $\lambda$ (n) increases to obtain the desired value of ARL₀. For example, when $n=5$ and ${ARL}_0\approx 370$, the $L_{DH}$ values are equal to 1.000, 1.280, 1.356, 1.492, 1.688, 1.915 and 2.027 for $\lambda =0.10,0.14,0.15,0.17,0.20,0.25$ and 0.30, respectively. Moreover, when $\lambda =0.15$ and ${ARL}_0\approx 500$, the $L_{DH}$ values are equal to 1.448, 1.598, 1.631 and 1.650 for $n=5,10,15$ and 20, respectively.

4.2. IC Robustness

As the DHWMA-SR chart is nonparametric, its IC run-length characteristics should be approximately the same for any continuous and symmetric distribution. To study the IC robustness of the proposed chart, we use the following distributions:

• The standard normal distribution, $N(0,1)$;
• The scaled Student’s t-distribution, $t\left(\nu \right)/\sqrt{\nu /(\nu -2)}$ with degrees of freedom $\nu =4$ and 8;
• The logistic distribution, $LG(0,\sqrt{3}/\pi )$;
• The double exponential or Laplace distribution, $L(0,1/\sqrt{2})$;
• The contaminated normal (CN) distribution, which is a mixture of $N(0,{\sigma }_1^2)$ and $N(0,{\sigma }_2^2)$, represented by $(1-\alpha )N(0,{\sigma }_1^2)+\alpha N(0,{\sigma }_2^2)$ with ${\sigma }_1/{\sigma }_2=2$ and level of contamination $\alpha =0.05$;
• The uniform distribution, $U(-\sqrt{3},\sqrt{3})$.

The scaled Student’s t and the logistic distributions have heavier tails than the normal one, while the Laplace distribution has fatter tails than the normal. On the other hand, the CN distribution is used to study the effects of outliers. Note that all distributions have mean/median equal to 0 and standard deviation of 1.

Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12, Figure A13, Figure A14 in the Appendix B show the IC run-length characteristics ARL₀, MDRL₀ and SDRL₀ of the DHWMA-SR chart under the above distributions for $n=5$ and 10 respectively, given an $ARL0\approx 370$. From these figures, we observe that for a specified value of $\lambda$, the IC run-length characteristics remain approximately stable across all distributions. Comparing the Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12, Figure A13, Figure A14 it is seen that for a specified value of $\lambda$, the MDRL₀ (SDRL₀) values for $n=5$ are smaller (larger) than the corresponding MDRL₀ (SDRL₀) values for $n=10$. For example, when $\lambda =0.15$, the MDRL₀ and SDRL₀ values of the DHWMA-SR chart for $n=5$ are 90 and 537, while the corresponding values for $n=10$ are 183 and 427. It is also important to mention that for a specified value of n, the MDRL₀ (SDRL₀) value increases up (decreases down) to the value of $\lambda =0.25$ and subsequently, it decreases (increases). Moreover, for $n=5$, the SDRL₀ value ranges from 859 (when $\lambda =0.10$) to 430 (when $\lambda =0.25$), while for $n=10$, the SDRL₀ value ranges from 698 (when $\lambda =0.10$) to 284 (when $\lambda =0.25$). From these findings, we observe that the IC run-length distribution is more positively-skewed for small values of n and $\lambda$. According to Chan and Zhang [33], a large value of SDRL₀ may lead to a high probability of an OOC signal. For this reason, we recommend practitioners to use values $\lambda$ = 0.20–0.25 for the proposed scheme when $n=5$ and $\lambda \ge 0.15$ when $n\in [10,20]$.

4.3. OOC Performance

To study the OOC performance of the DHWMA-SR chart, we use the combinations of design parameters ($\lambda ,L_{DH}$) shown in Table 1. We also consider the positive shifts $\delta =0.05,0.10,0.25,0.50,0.75,1.00,1.25,1.50$ in terms of population standard deviation. The results are similar for negative shifts as the underlying distribution is symmetric around the mean/median.

Table 2. Zero-state run-length characteristics for the DHWMA-SR chart under various distributions when $AR{L}_0\approx 370$ and $n=5$.

				$\mathit{\delta }$
$\mathit{\lambda }$	${\mathit{L}}_{\mathit{DH}}$	Distribution	Characteristic	0	0.05	0.10	0.25	0.50	0.75	1.00	1.25	1.50
0.10	1.000	$N(0,1)$	ARL	370.56	91.48	35.62	9.13	3.37	1.95	1.36	1.12	1.03
			MDRL	9	8	7	4	1	1	1	1	1
			SDRL	850.96	179.61	60.58	12.16	3.53	1.78	1.02	0.55	0.27
		$t_4$	ARL	376.39	69.10	26.26	6.81	2.67	1.66	1.28	1.13	1.06
			MDRL	10	8	7	4	1	1	1	1	1
			SDRL	856.20	129.31	42.47	8.44	2.63	1.41	0.85	0.54	0.36
		$t_8$	ARL	371.82	82.46	32.05	8.28	3.14	1.85	1.34	1.13	1.05
			MDRL	9	8	7	4	1	1	1	1	1
			SDRL	852.12	159.24	53.73	10.81	3.21	1.65	0.95	0.56	0.33
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	371.14	80.66	31.58	8.14	3.08	1.84	1.34	1.13	1.05
			MDRL	9	8	7	4	1	1	1	1	1
			SDRL	848.03	155.58	52.77	10.55	3.17	1.64	0.96	0.56	0.33
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	374.27	60.82	23.18	6.40	2.69	1.73	1.33	1.15	1.07
			MDRL	10	8	6	4	1	1	1	1	1
			SDRL	858.95	110.50	36.85	7.71	2.61	1.47	0.93	0.59	0.39
		$CN(a=0.05)$	ARL	371.33	88.43	34.20	8.76	3.25	1.88	1.32	1.10	1.03
			MDRL	9	9	7	4	1	1	1	1	1
			SDRL	852.31	170.91	58.14	11.57	3.36	1.70	0.94	0.50	0.25
		$U(-\sqrt{3},\sqrt{3})$	ARL	371.14	97.62	39.40	10.44	3.85	2.21	1.48	1.14	1.01
			MDRL	9	9	7	4	1	1	1	1	1
			SDRL	848.03	193.14	68.72	14.51	4.31	2.16	1.23	0.62	0.19
0.14	1.280	$N(0,1)$	ARL	370.53	127.75	52.40	12.87	4.35	2.39	1.60	1.25	1.09
			MDRL	65	41	23	7	3	1	1	1	1
			SDRL	571.41	187.26	69.99	14.73	4.16	2.05	1.21	0.73	0.41
		$t_4$	ARL	372.31	98.82	38.19	9.21	3.31	1.96	1.45	1.23	1.12
			MDRL	65	36	18	6	2	1	1	1	1
			SDRL	572.09	140.50	49.38	10.00	3.03	1.63	1.04	0.72	0.52
		$t_8$	ARL	372.60	116.32	46.46	11.32	3.91	2.22	1.55	1.25	1.11
			MDRL	66	39	21	7	3	1	1	1	1
			SDRL	572.98	169.02	61.60	12.69	3.66	1.90	1.17	0.75	0.49
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	372.32	117.18	46.21	11.32	3.92	2.22	1.56	1.25	1.11
			MDRL	67	40	21	7	3	1	1	1	1
			SDRL	572.96	169.10	60.91	12.59	3.66	1.88	1.16	0.75	0.48
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	372.32	87.54	33.71	8.59	3.32	2.05	1.52	1.26	1.14
			MDRL	67	33	16	5	2	1	1	1	1
			SDRL	572.96	121.85	42.89	9.15	3.02	1.71	1.13	0.78	0.55
		$CN(a=0.05)$	ARL	366.05	124.46	50.14	12.14	4.12	2.28	1.55	1.22	1.08
			MDRL	64	41	22	7	3	1	1	1	1
			SDRL	565.39	180.90	66.80	13.72	3.89	1.95	1.15	0.67	0.37
		$U(-\sqrt{3},\sqrt{3})$	ARL	372.32	139.37	58.13	14.85	5.11	2.77	1.78	1.28	1.05
			MDRL	67	44	24	8	4	2	1	1	1
			SDRL	572.96	205.38	78.34	17.30	5.06	2.47	1.43	0.78	0.30
0.15	1.356	$N(0,1)$	ARL	369.85	134.33	55.66	13.64	4.56	2.49	1.65	1.27	1.10
			MDRL	88	51	27	8	3	1	1	1	1
			SDRL	535.34	187.12	71.12	15.18	4.29	2.13	1.27	0.77	0.43
		$t_4$	ARL	372.55	104.13	40.59	9.73	3.45	2.02	1.48	1.24	1.12
			MDRL	89	44	21	6	2	1	1	1	1
			SDRL	536.54	141.08	50.33	10.32	3.12	1.69	1.08	0.74	0.53
		$t_8$	ARL	371.81	122.31	49.36	11.97	4.09	2.30	1.59	1.26	1.12
			MDRL	90	48	24	7	3	1	1	1	1
			SDRL	536.39	169.13	62.79	13.10	3.78	1.96	1.22	0.78	0.50
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	372.09	123.46	49.16	11.97	4.10	2.30	1.59	1.26	1.11
			MDRL	90	49	25	7	3	1	1	1	1
			SDRL	537.41	169.68	62.09	12.99	3.78	1.95	1.21	0.78	0.49
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	372.09	92.54	35.83	9.05	3.46	2.11	1.55	1.28	1.14
			MDRL	90	41	19	6	2	1	1	1	1
			SDRL	537.41	123.01	43.83	9.45	3.11	1.77	1.17	0.81	0.56
		$CN(a=0.05)$	ARL	367.09	130.51	53.21	12.86	4.32	2.37	1.59	1.23	1.08
			MDRL	87	50	26	8	3	1	1	1	1
			SDRL	532.25	181.00	67.80	14.13	4.02	2.02	1.20	0.70	0.39
		$U(-\sqrt{3},\sqrt{3})$	ARL	372.09	146.55	61.67	15.75	5.39	2.92	1.86	1.31	1.06
			MDRL	90	55	29	9	4	2	1	1	1
			SDRL	537.41	205.34	79.47	23.00	5.22	2.56	1.50	0.83	0.31
0.17	1.492	$N(0,1)$	ARL	370.26	146.89	62.38	15.44	5.10	2.79	1.87	1.42	1.18
			MDRL	146	76	37	10	4	2	1	1	1
			SDRL	471.94	184.05	71.93	15.83	4.50	2.21	1.36	0.89	0.57
		$t_4$	ARL	372.64	114.50	45.69	10.98	3.83	2.24	1.63	1.34	1.18
			MDRL	148	63	28	8	3	1	1	1	1
			SDRL	473.84	139.61	51.29	10.84	3.26	1.76	1.17	0.84	0.62
		$t_8$	ARL	373.53	134.57	55.39	13.53	4.57	2.57	1.78	1.39	1.20
			MDRL	150	71	33	9	4	2	1	1	1
			SDRL	474.77	166.77	63.69	13.69	3.96	2.04	1.31	0.89	0.62
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	372.67	135.14	55.40	13.50	4.58	2.57	1.78	1.40	1.20
			MDRL	148	72	34	9	4	2	1	1	1
			SDRL	475.41	166.43	63.20	13.57	3.95	2.03	1.31	0.89	0.61
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	372.67	102.46	40.36	10.15	3.82	2.33	1.71	1.39	1.21
			MDRL	148	58	26	7	3	1	1	1	1
			SDRL	475.41	122.54	44.82	9.89	3.25	1.85	1.26	0.91	0.66
		$CN(a=0.05)$	ARL	367.92	143.13	59.55	14.56	4.82	2.66	1.80	1.37	1.16
			MDRL	144	75	35	10	4	2	1	1	1
			SDRL	470.68	178.06	68.68	14.76	4.20	2.10	1.30	0.84	0.53
		$U(-\sqrt{3},\sqrt{3})$	ARL	372.67	159.37	69.20	17.88	6.08	3.31	2.13	1.49	1.12
			MDRL	148	81	41	12	5	3	1	1	1
			SDRL	475.41	199.96	80.27	18.55	5.46	2.65	1.58	0.97	0.46
0.20	1.688	$N(0,1)$	ARL	369.35	154.45	67.33	17.12	5.60	3.01	1.97	1.46	1.20
			MDRL	192	95	46	12	4	3	1	1	1
			SDRL	431.92	177.01	71.30	16.39	4.78	2.38	1.48	0.97	0.61
		$t_4$	ARL	368.68	121.75	49.80	12.16	4.17	2.38	1.69	1.36	1.19
			MDRL	190	78	35	9	4	1	1	1	1
			SDRL	431.47	135.78	51.46	11.33	3.47	1.89	1.26	0.89	0.65
		$t_8$	ARL	371.03	142.12	60.06	14.99	5.00	2.76	1.86	1.43	1.21
			MDRL	193	89	41	11	4	2	1	1	1
			SDRL	433.32	160.94	63.39	14.20	4.21	2.19	1.41	0.96	0.65
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	370.26	142.47	60.19	14.98	5.00	2.75	1.86	1.43	1.21
			MDRL	192	89	42	11	4	2	1	1	1
			SDRL	433.44	160.74	63.01	14.10	4.20	2.18	1.41	0.96	0.65
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	370.26	109.03	44.07	11.23	4.15	2.46	1.77	1.42	1.23
			MDRL	192	71	31	8	4	1	1	1	1
			SDRL	433.44	119.58	44.98	10.35	3.47	1.99	1.36	0.96	0.69
		$CN(a=0.05)$	ARL	367.29	150.32	64.60	16.17	5.28	2.86	1.89	1.41	1.17
			MDRL	188	93	44	12	4	2	1	1	1
			SDRL	430.40	171.00	68.38	15.35	4.45	2.56	1.41	0.90	0.56
		$U(-\sqrt{3},\sqrt{3})$	ARL	370.26	167.51	74.95	19.95	6.75	3.62	2.27	1.53	1.12
			MDRL	192	102	51	15	5	3	1	1	1
			SDRL	433.44	192.48	79.75	19.24	5.78	2.84	1.71	1.04	0.48
0.25	1.915	$N(0,1)$	ARL	370.45	155.41	70.30	18.92	6.36	3.49	2.35	1.77	1.43
			MDRL	207	106	53	15	5	3	2	1	1
			SDRL	429.72	162.85	66.38	16.00	4.78	2.37	1.51	1.08	0.78
		$t_4$	ARL	369.94	123.56	52.52	13.53	4.75	2.77	2.00	1.60	1.37
			MDRL	204	87	40	11	4	2	1	1	1
			SDRL	429.27	125.04	48.19	11.14	3.48	1.91	1.33	1.01	0.79
		$t_8$	ARL	370.68	143.49	63.25	16.65	5.68	3.19	2.21	1.71	1.42
			MDRL	206	99	48	13	5	3	2	1	1
			SDRL	430.15	148.82	59.33	13.94	4.22	2.19	1.46	1.07	0.80
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	370.80	144.03	63.25	16.59	5.68	3.20	2.22	1.71	1.42
			MDRL	204	100	48	13	5	3	2	1	1
			SDRL	431.87	148.19	58.98	13.84	4.21	2.18	1.46	1.07	0.80
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	370.80	111.17	46.92	12.49	4.70	2.85	2.09	1.68	1.43
			MDRL	204	80	37	10	4	2	1	1	1
			SDRL	431.87	110.18	42.52	10.23	3.48	2.00	1.42	1.07	0.84
		$CN(a=0.05)$	ARL	368.62	151.42	67.56	17.86	6.02	3.33	2.26	1.71	1.39
			MDRL	203	104	51	14	5	3	2	1	1
			SDRL	428.99	158.06	63.71	15.01	4.46	2.25	1.45	1.02	0.73
		$U(-\sqrt{3},\sqrt{3})$	ARL	370.80	167.77	78.23	22.05	7.70	4.20	2.75	1.93	1.38
			MDRL	204	113	58	18	6	4	2	2	1
			SDRL	431.87	177.42	74.56	18.74	5.81	2.82	1.70	1.11	0.66
0.30	2.027	$N(0,1)$	ARL	368.72	141.80	66.17	18.88	6.65	3.84	2.82	2.37	2.15
			MDRL	170	96	50	15	6	3	2	2	2
			SDRL	549.86	148.36	59.39	14.70	4.39	2.03	1.14	0.69	0.42
		$t_4$	ARL	368.17	113.50	50.20	13.66	5.08	3.21	2.56	2.28	2.14
			MDRL	170	80	39	11	4	3	2	2	2
			SDRL	543.16	112.47	43.47	10.24	3.12	1.53	0.94	0.63	0.44
		$t_8$	ARL	371.08	131.24	59.93	16.68	5.99	3.59	2.72	2.34	2.16
			MDRL	171	90	46	14	5	3	2	2	2
			SDRL	548.95	134.54	53.25	12.79	3.85	1.83	1.07	0.68	0.45
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	368.96	131.62	59.84	16.62	5.98	3.58	2.72	2.34	2.16
			MDRL	168	90	46	14	5	3	2	2	2
			SDRL	550.61	134.20	53.08	12.67	3.82	1.83	1.07	0.68	0.45
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	368.96	102.57	44.87	12.65	5.05	3.29	2.63	2.33	2.17
			MDRL	168	74	36	10	4	3	2	2	2
			SDRL	550.61	99.24	38.22	9.37	3.11	1.62	1.02	0.69	0.48
		$CN(a=0.05)$	ARL	364.59	138.32	63.72	17.86	6.32	3.69	2.74	2.32	2.13
			MDRL	169	94	48	15	5	3	2	2	2
			SDRL	537.41	143.54	57.04	13.77	4.10	1.91	1.07	0.65	0.39
		$U(-\sqrt{3},\sqrt{3})$	ARL	368.96	152.86	73.40	21.97	7.98	4.51	3.13	2.45	2.10
			MDRL	168	101	56	18	7	4	3	2	2
			SDRL	550.61	162.83	66.92	17.23	5.40	2.52	1.37	0.77	0.33

and

Table 3. Zero-state run-length characteristics for the DHWMA-SR chart under various distributions when $AR{L}_0\approx 370$ and $n=10$.

				$\mathit{\delta }$
$\mathit{\lambda }$	${\mathit{L}}_{\mathit{DH}}$	Distribution	Characteristic	0	0.05	0.10	0.25	0.50	0.75	1.00	1.25	1.50
0.10	1.071	$N(0,1)$	ARL	371.65	66.86	23.90	5.86	2.16	1.30	1.06	1.01	1.00
			MDRL	17	13	8	4	1	1	1	1	1
			SDRL	696.43	112.83	35.33	6.64	1.97	0.89	0.37	0.13	0.04
		$t_4$	ARL	371.40	49.72	17.67	4.41	1.75	1.18	1.04	1.01	1.00
			MDRL	17	11	7	3	1	1	1	1	1
			SDRL	695.73	79.93	24.84	4.68	1.48	0.65	0.28	0.13	0.07
		$t_8$	ARL	366.02	59.79	21.38	5.27	2.01	1.25	1.05	1.01	1.00
			MDRL	16	12	8	3	1	1	1	1	1
			SDRL	690.48	99.41	31.17	5.82	1.80	0.81	0.34	0.13	0.05
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	366.63	60.15	21.44	5.31	2.01	1.25	1.05	1.01	1.00
			MDRL	17	12	8	4	1	1	1	1	1
			SDRL	693.01	99.38	31.17	5.85	1.78	0.79	0.33	0.13	0.05
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	366.63	45.56	16.11	4.26	1.77	1.20	1.05	1.01	1.00
			MDRL	17	11	7	3	1	1	1	1	1
			SDRL	693.01	72.66	22.37	4.47	1.50	0.70	0.33	0.15	0.07
		$CN(a=0.05)$	ARL	374.55	64.28	22.91	5.61	2.06	1.26	1.05	1.01	1.00
			MDRL	17	12	8	4	1	1	1	1	1
			SDRL	698.97	107.41	33.52	6.29	1.86	0.82	0.33	0.11	0.03
		$U(-\sqrt{3},\sqrt{3})$	ARL	369.00	69.28	25.22	6.38	2.41	1.42	1.09	1.01	1.00
			MDRL	17	13	8	4	1	1	1	1	1
			SDRL	693.71	116.62	37.43	7.47	2.29	1.09	0.48	0.14	0.01
0.14	1.395	$N(0,1)$	ARL	370.16	96.72	35.26	7.94	2.68	1.50	1.12	1.02	1.00
			MDRL	139	48	20	5	1	1	1	1	1
			SDRL	462.14	122.89	41.38	7.94	2.29	1.09	0.51	0.20	0.07
		$t_4$	ARL	369.52	73.00	25.68	5.84	2.11	1.31	1.08	1.02	1.01
			MDRL	139	39	15	4	1	1	1	1	1
			SDRL	462.17	90.24	29.35	5.55	1.73	0.83	0.40	0.20	0.10
		$t_8$	ARL	368.50	87.93	31.49	7.08	2.46	1.43	1.11	1.02	1.00
			MDRL	137	44	18	5	1	1	1	1	1
			SDRL	461.36	110.47	36.85	6.97	2.08	1.00	0.47	0.20	0.09
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	369.43	88.01	31.51	7.12	2.47	1.43	1.11	1.02	1.00
			MDRL	137	45	18	5	1	1	1	1	1
			SDRL	462.20	109.98	36.64	6.97	2.06	0.99	0.47	0.20	0.08
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	369.43	66.98	23.45	5.59	2.14	1.35	1.10	1.03	1.01
			MDRL	137	36	14	4	1	1	1	1	1
			SDRL	462.20	82.26	26.55	5.27	1.75	0.89	0.45	0.23	0.11
		$CN(a=0.05)$	ARL	371.38	93.37	33.81	7.59	2.56	1.44	1.10	1.02	1.00
			MDRL	140	47	20	5	1	1	1	1	1
			SDRL	464.13	118.11	39.48	7.52	2.16	1.01	0.45	0.17	0.05
		$U(-\sqrt{3},\sqrt{3})$	ARL	369.43	101.30	37.80	8.82	3.05	1.69	1.18	1.02	1.00
			MDRL	137	50	21	6	2	1	1	1	1
			SDRL	462.20	128.06	44.70	8.96	2.67	1.31	0.61	0.21	0.03
0.15	1.479	$N(0,1)$	ARL	370.63	103.71	38.14	8.53	2.85	1.58	1.15	1.03	1.00
			MDRL	183	59	24	6	2	1	1	1	1
			SDRL	427.32	123.66	42.36	8.20	2.36	1.15	0.55	0.22	0.08
		$t_4$	ARL	369.30	78.42	27.77	6.24	2.23	1.36	1.10	1.03	1.01
			MDRL	182	47	18	5	1	1	1	1	1
			SDRL	427.05	91.21	30.16	5.74	1.79	0.88	0.44	0.22	0.12
		$t_8$	ARL	369.79	94.36	34.03	7.59	2.61	1.49	1.13	1.03	1.01
			MDRL	183	54	21	5	1	1	1	1	1
			SDRL	425.88	111.40	37.71	7.21	2.15	1.05	>0.51	0.23	0.10
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	370.47	94.37	34.05	7.62	2.61	1.50	1.13	1.03	1.01
			MDRL	183	55	22	6	1	1	1	1	1
			SDRL	427.02	110.77	37.45	7.20	2.13	1.04	0.51	0.23	0.09
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	370.47	72.05	25.32	5.96	2.26	1.40	1.12	1.04	1.01
			MDRL	183	43	16	5	1	1	1	1	1
			SDRL	427.02	83.16	27.22	5.44	1.80	0.94	0.49	0.26	0.13
		$CN(a=0.05)$	ARL	370.63	100.05	36.54	8.12	2.71	1.51	1.12	1.02	1.00
			MDRL	184	57	23	6	2	1	1	1	1
			SDRL	427.00	118.69	40.40	7.78	2.22	1.07	0.49	0.19	0.06
		$U(-\sqrt{3},\sqrt{3})$	ARL	370.47	108.54	40.87	9.49	3.25	1.78	1.21	1.03	1.00
			MDRL	183	62	25	7	2	1	1	1	1
			SDRL	427.02	128.51	45.66	9.27	2.75	1.36	0.66	0.23	0.04
0.17	1.650	$N(0,1)$	ARL	369.54	114.66	43.04	9.60	3.15	1.71	1.20	1.04	1.01
			MDRL	251	77	30	7	3	1	1	1	1
			SDRL	373.04	123.00	43.68	8.69	2.51	1.25	0.63	0.28	0.10
		$t_4$	ARL	368.93	87.50	31.40	7.00	2.45	1.44	1.13	1.04	1.01
			MDRL	249	60	22	5	1	1	1	1	1
			SDRL	372.71	91.98	31.25	6.10	1.91	0.97	0.51	0.26	0.14
		$t_8$	ARL	368.52	104.78	38.46	8.58	2.87	1.61	1.18	1.04	1.01
			MDRL	249	71	27	6	2	1	1	1	1
			SDRL	372.20	111.52	38.94	7.67	2.28	1.15	0.59	0.28	0.12
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	369.23	104.70	38.57	8.57	2.87	1.61	1.18	1.04	1.01
			MDRL	249	71	27	7	2	1	1	1	1
			SDRL	372.14	110.84	38.78	7.64	2.26	1.14	0.59	0.28	0.13
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	369.23	80.53	28.74	6.65	2.47	1.50	1.17	1.05	1.01
			MDRL	249	56	21	5	1	1	1	1	1
			SDRL	372.14	84.02	28.38	5.77	1.93	1.03	0.57	0.31	0.16
		$CN(a=0.05)$	ARL	368.64	110.57	41.24	9.15	2.99	1.63	1.16	1.03	1.00
			MDRL	250	74	29	7	2	1	1	1	1
			SDRL	372.45	118.14	41.61	8.23	2.36	1.17	0.57	0.24	0.08
		$U(-\sqrt{3},\sqrt{3})$	ARL	369.23	120.34	46.21	10.74	3.61	1.94	1.28	1.04	1.00
			MDRL	249	81	33	8	3	1	1	1	1
			SDRL	372.14	128.22	46.93	9.82	2.92	1.47	0.76	0.29	0.05
0.20	1.890	$N(0,1)$	ARL	370.28	127.97	49.69	11.26	3.69	2.01	1.35	1.10	1.02
			MDRL	305	98	39	9	3	1	1	1	1
			SDRL	317.69	119.61	44.14	9.15	2.62	1.36	0.76	0.38	0.17
		$t_4$	ARL	367.73	98.61	36.48	8.19	2.87	1.69	1.26	1.11	1.05
			MDRL	302	76	29	7	3	1	1	1	1
			SDRL	317.57	90.73	31.98	6.43	1.99	1.09	0.64	0.39	0.25
		$t_8$	ARL	368.20	117.56	44.58	10.03	3.36	1.88	1.32	1.11	1.03
			MDRL	304	90	35	8	3	1	1	1	1
			SDRL	316.61	109.37	39.60	8.08	2.37	1.26	0.72	0.40	0.21
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	369.19	117.27	44.55	10.05	3.36	1.89	1.33	1.11	1.03
			MDRL	305	90	35	8	3	1	1	1	1
			SDRL	317.09	108.22	39.29	8.03	2.35	1.25	0.72	0.39	0.21
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	369.19	91.00	33.39	7.78	2.90	1.76	1.32	1.13	1.05
			MDRL	305	71	26	6	3	1	1	1	1
			SDRL	317.09	82.82	29.08	6.09	2.01	1.14	0.71	0.43	0.26
		$CN(a=0.05)$	ARL	368.84	123.39	47.65	10.74	3.51	1.91	1.30	1.08	1.02
			MDRL	302	94	37	9	3	1	1	1	1
			SDRL	317.49	115.01	42.21	8.66	2.46	1.28	0.70	0.34	0.14
		$U(-\sqrt{3},\sqrt{3})$	ARL	369.19	133.80	53.34	12.65	4.24	2.29	1.47	1.11	1.01
			MDRL	305	102	42	10	4	2	1	1	1
			SDRL	317.09	124.28	47.58	10.33	3.05	1.56	0.89	0.40	0.10
0.25	2.229	$N(0,1)$	ARL	369.93	138.24	55.72	13.25	4.40	2.45	1.63	1.24	1.07
			MDRL	321	112	46	11	4	2	1	1	1
			SDRL	284.58	114.78	43.47	9.40	2.70	1.43	0.92	0.56	0.31
		$t_4$	ARL	367.49	107.89	41.34	9.66	3.42	2.04	1.48	1.23	1.12
			MDRL	318	88	35	8	3	2	1	1	1
			SDRL	284.09	87.80	31.74	6.66	2.06	1.21	0.82	0.58	0.41
		$t_8$	ARL	367.34	127.27	50.25	11.80	4.00	2.28	1.58	1.25	1.10
			MDRL	319	103	42	10	4	2	1	1	1
			SDRL	283.47	105.02	39.10	8.33	2.45	1.35	0.89	0.58	0.36
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	368.50	127.15	50.21	11.83	4.01	2.29	1.58	1.25	1.10
			MDRL	320	103	42	10	4	2	1	1	1
			SDRL	284.04	104.56	38.76	8.29	2.43	1.35	0.89	0.58	0.36
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	368.50	99.62	37.85	9.18	3.45	2.12	1.55	1.27	1.13
			MDRL	320	82	32	8	3	2	1	1	1
			SDRL	284.04	80.45	28.89	6.30	2.09	1.26	0.88	0.62	0.43
		$CN(a=0.05)$	ARL	368.11	133.51	53.50	12.61	4.19	2.33	1.56	1.20	1.06
			MDRL	319	108	45	11	4	2	1	1	1
			SDRL	284.11	110.70	41.69	8.90	2.54	1.36	0.87	0.52	0.27
		$U(-\sqrt{3},\sqrt{3})$	ARL	368.50	144.72	59.94	14.90	5.07	2.80	1.81	1.26	1.03
			MDRL	320	116	50	13	5	3	1	1	1
			SDRL	284.04	119.68	46.90	10.61	3.14	1.61	1.03	0.58	0.20
0.30	2.451	$N(0,1)$	ARL	370.61	140.16	57.19	14.02	4.76	2.70	1.85	1.39	1.15
			MDRL	309	113	48	12	4	3	2	1	1
			SDRL	291.90	114.52	42.65	9.20	2.65	1.39	0.95	0.66	0.41
		$t_4$	ARL	368.00	109.39	42.56	10.29	3.71	2.25	1.64	1.33	1.18
			MDRL	308	89	36	9	3	2	1	1	1
			SDRL	290.76	86.65	30.87	6.56	2.01	1.21	0.88	0.65	0.49
		$t_8$	ARL	368.29	129.17	51.57	12.52	4.33	2.52	1.77	1.38	1.17
			MDRL	310	104	43	11	4	2	1	1	1
			SDRL	289.68	104.20	38.20	8.17	2.40	1.33	0.93	0.66	0.46
		$LG(0,{\displaystyle \frac{\sqrt{3}}{\pi }})$	ARL	368.43	128.73	51.46	12.53	4.34	2.53	1.78	1.38	1.17
			MDRL	311	104	43	11	4	3	1	1	1
			SDRL	289.31	103.59	37.91	8.11	2.39	1.32	0.93	0.66	0.45
		$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	368.43	101.05	39.00	9.76	3.73	2.32	1.72	1.39	1.20
			MDRL	311	83	33	9	4	2	1	1	1
			SDRL	289.31	79.36	28.11	6.20	2.06	1.26	0.93	0.69	0.51
		$CN(a=0.05)$	ARL	368.93	135.37	54.84	13.35	4.54	2.59	1.77	1.34	1.13
			MDRL	308	108	46	12	4	3	1	1	1
			SDRL	290.93	110.26	40.79	8.73	2.49	1.33	0.91	0.61	0.37
		$U(-\sqrt{3},\sqrt{3})$	ARL	368.43	147.00	61.58	15.78	5.48	3.10	2.05	1.42	1.07
			MDRL	311	118	51	14	5	3	2	1	1
			SDRL	289.31	119.83	46.17	10.43	3.08	1.55	1.03	0.67	0.28

show the zero-state run-length characteristics (ARL, MDRL and SDRL) of the DHWMA-SR chart for $n=5$ and 10, respectively, given an $AR{L}_0\approx 370$. From these tables, we observe the following:

• The DHWMA-SR chart performs better as the value of n increases. For example, when $n=5$ and $\lambda =0.15$, the ARL₁ value of the proposed chart is equal to 40.59 at $\delta =0.10$ under the $t_4$ distribution, while the corresponding value is equal to 27.77 for $n=10$.
• The detection ability of the proposed chart is better as the value of $\lambda$ decreases. For example, when $n=10$ and $\delta =0.05$, the ARL₁ values of the DHWMA-SR chart under the $N(0,1)$ distribution are equal to 66.86, 96.72, 103.71, 114.66, 127.97, 138.24 and 140.16 for $\lambda =0.10,0.14,0.15,0.17,0.20,0.25$ and 0.30, respectively. However, we remind that this superiority of the DHWMA-SR chart is accompanied by a larger value of SDRL₀.
• For small to moderate shifts ($\delta \le 0.50$), the proposed chart performs better when the underlying distribution is the $L(0,1/\sqrt{2})$, while for larger shifts ($\delta >0.50$), it performs better when the underlying distribution is the $t_4$. On the other hand, the worst performance is observed under the $U(-\sqrt{3},\sqrt{3})$ for $\delta \le 1.25$ and under the $L(0,1/\sqrt{2})$ for $\delta =1.50$.
• Similar to the IC run-length distribution, the OOC run-length distribution is also positively skewed as $AR{L}_1>MDR{L}_1$.

5. Performance Analysis for the Steady-State Scenario

The results reported in Table 2 and Table 3 represent the zero-state performance of the DHWMA-SR chart, as it is assumed that the shift in the location parameter occurs in the beginning. In this subsection, we investigate the OOC performance of the proposed chart when the process is IC for a long time period and the shift occurs later; this case is referred to as steady-state. In other words, under the steady-state scenario, $\theta ={\theta }_0$ for $t<\tau$ and $\theta \ne {\theta }_0$ for $t\ge \tau$, where $\tau$ denotes the unknown change point.

Before we present the steady-state behavior of the DHWMA-SR chart, we discuss the steady-state performance of the HWMA scheme. Knoth et al. [23] pointed out that the HWMA scheme does not perform well in both the zero- and steady-state cases compared to the EWMA scheme, while Riaz et al. [34] showed that an HWMA scheme with suitable design parameters may be superior to an EWMA chart under the zero-state case, while it is more effective than the EWMA chart for large shifts under the steady-state case. However, both of these works do not provide a fair comparison study, as the ARL₀ values for the steady-state case are not computed. At this point, we note that the ARL₀ value of an EWMA chart with time-varying control limits converges quickly to a specific value as the value of $\tau$ increases, while this is not true for the HWMA chart. More specifically, the ARL₀ values of the HWMA chart change significantly over the time, especially for $\lambda \le 0.10$. In Appendix A,

Table A1. IC steady-state run-length characteristics for the HWMA-SR chart for different valious of $\tau$.

			$\mathit{n}=5$						$\mathit{n}=10$
$\mathit{\lambda }$	${\mathit{L}}_{\mathit{H}}$	Characteristic	$\mathit{\tau }=\mathbf{1}$	100	200	500	1000	${\mathit{L}}_{\mathit{H}}$	$\mathit{\tau }=\mathbf{1}$	100	200	500	1000
0.05	2.070	ARL	370.48	393.87	382.08	363.12	345.08	2.308	370.74	345.11	300.33	202.37	124.89
		MDRL	188	205	181	122	66		295	274	223	121	68
		SDRL	525.06	542.16	557.24	590.75	629.56		322.32	303.07	285.06	221.42	148.02
0.10	2.165	ARL	368.91	553.66	678.70	1119.37	1915.95	2.588	370.18	334.65	303.40	252.76	222.43
		MDRL	127	116	111	119	182		282	246	211	165	145
		SDRL	809.00	1628.41	1897.00	2593.39	3435.38		329.79	320.81	304.82	264.76	234.81
0.25	2.113	ARL	370.09	1387.79	2380.22	5626.89	9952.47	2.701	372.13	374.90	370.40	366.23	362.50
		MDRL	50	57	70	186	2603		256	261	101	254	253
		SDRL	1588.98	4798.64	6219.96	8771.89	9829.65		377.96	376.66	372.54	364.23	359.17

presents the IC run-length characteristics of the HWMA-SR chart for several values of $\lambda$, n and $\tau$. It is clearly seen that for $n=5$, the ARL₀ values increase as the value of $\tau$ also increases, while the corresponding SDRL₀ values are too large. On the other side, for $n=10$, the ARL₀ values decrease significantly as the value of $\tau$ increases for $\lambda =0.05$ and 0.10, while the steady-state ARL₀ values of the HWMA-SR chart with $\lambda =0.25$ are very close to the corresponding zero-state one for all of the considered values of $\tau$.

Table 4. IC steady-state run-length characteristics of the DHWMA-SR chart for different valious of $\tau$.

		$\mathit{n}=5$				$\mathit{n}=10$
$\mathit{\lambda }$	Characteristic	$\mathit{\tau }=\mathbf{100}$	200	500	1000	100	200	500	1000
0.10	ARL	696.63	718.10	702.15	575.00	608.95	612.22	535.18	370.55
	MDRL	138	136	43	9	204	198	77	16
	SDRL	1087.35	1095.80	1079.97	963.18	812.63	806.86	746.51	592.30
0.14	ARL	491.08	464.19	362.49	206.29	434.97	394.73	261.18	108.59
	MDRL	214	177	36	10	278	224	61	16
	SDRL	612.50	594.38	521.11	368.23	458.43	436.92	344.31	181.08
0.15	ARL	468.92	435.10	324.74	170.17	408.85	363.23	224.43	84.49
	MDRL	228	182	36	10	282	221	58	17
	SDRL	563.04	542.32	463.70	308.99	413.40	390.67	294.05	138.83
0.17	ARL	419.44	375.76	258.77	113.29	372.14	316.69	177.11	61.27
	MDRL	230	172	30	11	287	212	55	18
	SDRL	478.37	454.45	373.11	213.99	350.24	325.63	228.50	94.47
0.20	ARL	385.18	335.76	217.66	88.44	330.64	270.76	137.14	54.29
	MDRL	225	159	30	14	271	191	54	24
	SDRL	427.03	402.52	320.24	168.68	292.24	268.31	172.06	74.53
0.25	ARL	355.27	306.42	211.25	112.13	306.18	245.66	134.48	78.97
	MDRL	184	115	32	20	253	175	71	45
	SDRL	428.16	407.40	338.18	217.54	263.78	239.66	157.76	93.98
0.30	ARL	343.74	319.97	282.32	243.93	303.14	253.60	171.75	131.41
	MDRL	122	73	36	29	237	175	104	82
	SDRL	565.77	577.88	577.41	568.61	274.19	252.52	188.66	144.19

shows the IC steady-state run-length characteristics of the DHWMA-SR chart given a zero-state ARL₀ $\approx 370$, when the shift occurs at $\tau =100,200,500$ and 1000. Comparing to the results of Table 2 and Table 3 for $\delta =0$, we observe that the IC steady-state run-length characteristics are not close to the corresponding zero-state values. More specifically, the run-length characteristics decrease as the value of $\tau$ increases, while for $\lambda \le 0.20$ when $n=5$ or $\lambda \le 0.17$ when $n=10$ and values of $\tau$ are small, the ARL₀ and MDRL₀ values are larger than the corresponding zero-state values.

The steady-state run-length characteristics of the DHWMA-SR chart under various distributions for $\tau =100$ and 500 when $n=5$ and 10 are presented in

Table 5. Steady-state run-length characteristics for the DHWMA-SR chart under various distributions when $n=5$.

					$\mathit{\delta }$
$\mathit{\tau }$	$\mathit{\lambda }$	${\mathit{L}}_{\mathit{DH}}$	Distribution	Characteristic	0	0.05	0.10	0.25	0.50	0.75	1.00	1.25	1.50
100	0.20	1.688	$N(0,1)$	ARL	385.18	206.60	122.62	59.92	35.90	27.59	23.67	21.59	20.44
				MDRL	225	159	108	58	35	27	23	21	20
				SDRL	427.03	205.09	106.66	45.21	25.25	19.00	16.22	14.81	14.05
			$t_8$	ARL	386.61	195.09	115.36	56.28	34.03	26.50	23.05	21.24	20.26
				MDRL	226	151	103	55	34	26	23	21	20
				SDRL	427.71	190.66	99.13	42.24	23.91	18.32	15.87	14.63	13.97
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	387.88	165.17	96.62	48.77	31.19	25.39	22.65	21.21	20.39
				MDRL	230	136	89	48	31	25	22	21	20
				SDRL	426.07	153.90	79.69	35.79	21.79	17.50	15.56	14.57	14.03
			$U(-\sqrt{3},\sqrt{3})$	ARL	387.88	218.37	131.74	65.05	39.43	30.08	25.18	22.19	20.24
				MDRL	230	166	115	63	39	30	25	22	20
				SDRL	426.07	217.89	115.91	49.48	27.86	20.68	17.17	15.15	13.89
500	0.20	1.688	$N(0,1)$	ARL	217.66	145.33	99.39	60.85	45.49	39.09	35.01	32.30	30.65
				MDRL	30	30	31	32	33	31	28	26	24
				SDRL	320.24	197.17	121.58	64.68	44.05	36.89	33.11	30.77	29.38
			$t_8$	ARL	213.02	137.67	93.89	58.92	44.72	38.53	34.75	32.40	31.00
				MDRL	29	29	30	32	33	31	28	26	25
				SDRL	316.04	186.04	114.41	62.12	42.96	36.25	32.77	30.79	29.60
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	215.38	123.02	84.63	54.65	42.38	36.96	33.82	31.96	30.76
				MDRL	30	30	31	34	33	29	27	26	25
				SDRL	318.99	160.11	98.86	56.00	40.50	34.94	32.10	30.49	29.46
			$U(-\sqrt{3},\sqrt{3})$	ARL	215.38	149.78	104.30	63.44	47.47	41.24	36.83	33.29	30.54
				MDRL	30	30	30	31	33	32	30	27	24
				SDRL	318.99	205.81	129.72	68.64	46.52	38.88	34.51	31.53	29.26
100	0.25	1.915	$N(0,1)$	ARL	355.27	184.12	110.52	54.84	32.78	24.76	20.84	18.75	17.58
				MDRL	184	136	96	52	32	24	20	18	17
				SDRL	428.16	186.80	96.12	40.75	22.76	17.05	14.44	13.08	12.33
			$t_8$	ARL	352.16	173.58	103.95	51.62	30.98	23.69	20.25	18.46	17.49
				MDRL	180	129	90	49	30	23	19	18	17
				SDRL	428.65	172.91	89.18	38.15	21.56	16.41	14.12	12.95	12.32
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	354.27	147.44	87.60	44.57	28.19	22.47	19.81	18.37	17.56
				MDRL	183	118	79	43	27	22	19	18	17
				SDRL	428.13	138.96	71.88	32.21	19.59	15.60	13.80	12.84	12.32
			$U(-\sqrt{3},\sqrt{3})$	ARL	354.27	193.19	118.61	59.62	36.18	27.25	22.38	19.35	17.40
				MDRL	183	140	102	56	35	26	22	19	17
				SDRL	428.13	198.09	104.55	44.71	25.15	18.61	15.34	13.40	12.19
500	0.25	1.915	$N(0,1)$	ARL	211.25	129.77	88.21	56.22	42.41	34.75	29.66	26.70	25.05
				MDRL	32	32	32	33	32	27	23	21	19
				SDRL	338.18	179.93	106.37	57.54	40.41	33.19	28.67	26.08	24.64
			$t_8$	ARL	206.49	122.34	83.88	54.93	41.52	34.04	29.33	26.74	25.34
				MDRL	31	31	32	33	31	26	23	21	19
				SDRL	333.48	168.13	100.46	55.81	39.44	32.48	28.36	26.09	24.83
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	209.15	109.24	76.40	51.46	38.69	32.14	28.42	26.34	25.16
				MDRL	32	32	33	34	30	25	22	20	19
				SDRL	337.25	142.41	87.08	51.12	36.97	31.02	27.69	25.80	24.73
			$U(-\sqrt{3},\sqrt{3})$	ARL	209.15	134.36	92.53	58.24	44.32	37.26	31.79	27.67	24.88
				MDRL	32	32	32	32	32	29	25	21	19
				SDRL	337.25	189.30	113.83	60.59	42.32	35.04	30.32	26.82	24.43

and

Table 6. Steady-state run-length characteristics for the DHWMA-SR chart under various distributions when $n=10$.

					$\mathit{\delta }$
$\mathit{\tau }$	$\mathit{\lambda }$	${\mathit{L}}_{\mathit{DH}}$	Distribution	Characteristic	0	0.05	0.10	0.25	0.50	0.75	1.00	1.25	1.50
100	0.17	1.650	$N(0,1)$	ARL	372.14	183.27	106.81	50.60	29.47	22.11	18.69	16.93	16.02
				MDRL	287	160	102	51	30	22	18	17	16
				SDRL	350.24	159.46	83.38	35.64	20.02	15.05	12.84	11.73	11.17
			$t_8$	ARL	374.03	174.79	101.28	48.17	28.25	21.51	18.47	16.93	16.13
				MDRL	289	155	97	48	28	21	18	17	16
				SDRL	351.43	149.79	77.83	33.55	19.09	14.60	12.64	11.67	11.18
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	373.27	149.84	86.33	42.18	25.88	20.52	18.12	16.87	16.19
				MDRL	286	136	84	42	26	20	18	17	16
				SDRL	352.01	124.61	64.91	29.15	17.56	14.02	12.47	11.67	11.24
			$U(-\sqrt{3},\sqrt{3})$	ARL	373.27	189.05	111.53	54.18	32.01	23.89	19.77	17.35	15.92
				MDRL	286	165	106	54	32	24	20	17	16
				SDRL	352.01	164.62	87.25	38.01	21.57	16.07	13.41	11.91	11.06
500	0.17	1.650	$N(0,1)$	ARL	177.11	132.74	98.34	59.78	38.85	29.64	24.92	22.37	21.02
				MDRL	55	54	50	41	28	21	17	15	14
				SDRL	228.50	158.41	107.90	59.27	37.79	29.69	25.78	23.62	22.46
			$t_8$	ARL	175.05	130.34	95.74	57.50	37.23	28.69	24.47	22.27	21.09
				MDRL	55	54	51	40	27	20	17	15	14
				SDRL	225.72	153.88	103.65	56.54	36.23	28.84	25.31	23.44	22.43
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	177.17	120.06	86.13	51.33	33.65	27.01	23.75	21.99	21.01
				MDRL	55	53	48	36	24	19	16	15	14
				SDRL	227.99	138.17	91.17	50.22	33.28	27.54	24.76	23.24	22.37
			$U(-\sqrt{3},\sqrt{3})$	ARL	177.17	135.97	101.99	63.48	42.31	32.51	26.65	22.97	20.72
				MDRL	55	54	52	44	32	25	19	16	14
				SDRL	227.99	161.90	111.54	62.48	40.28	31.67	26.90	23.93	22.07
100	0.25	2.229	$N(0,1)$	ARL	306.18	178.71	108.49	50.10	26.77	18.75	15.12	13.29	12.35
				MDRL	253	159	102	49	26	18	15	13	12
				SDRL	263.78	141.28	76.42	31.94	17.10	12.44	10.30	9.29	8.75
			$t_8$	ARL	307.10	171.26	102.57	47.28	25.39	18.03	14.80	13.22	12.42
				MDRL	253	153	97	46	25	17	14	13	12
				SDRL	265.17	133.64	71.14	29.81	16.25	11.99	10.12	9.21	8.74
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	307.22	148.97	87.28	40.58	22.65	16.96	14.45	13.17	12.48
				MDRL	251	135	83	39	22	16	14	13	12
				SDRL	266.83	112.47	59.30	25.68	14.79	11.44	9.97	9.22	8.82
			$U(-\sqrt{3},\sqrt{3})$	ARL	307.22	184.71	113.80	54.21	29.87	20.87	16.32	13.75	12.24
				MDRL	251	164	107	53	29	20	16	13	12
				SDRL	266.83	146.00	80.21	34.16	18.61	13.46	10.91	9.47	8.65
500	0.25	2.229	$N(0,1)$	ARL	134.48	117.05	94.31	56.14	29.64	19.18	14.62	12.47	11.41
				MDRL	71	68	61	41	19	10	6	4	3
				SDRL	157.76	126.70	94.04	52.29	30.01	21.25	17.23	15.27	14.28
			$t_8$	ARL	135.00	115.73	92.36	53.12	27.54	18.07	14.10	12.31	11.43
				MDRL	71	68	61	38	17	9	6	4	3
				SDRL	158.04	124.42	90.92	49.55	28.21	20.22	16.71	15.05	14.23
			$L(0,{\displaystyle \frac{1}{\sqrt{2}}})$	ARL	135.19	109.50	83.35	44.59	23.25	16.33	13.47	12.10	11.37
				MDRL	71	66	56	31	13	7	5	4	3
				SDRL	157.69	114.83	80.61	42.89	24.99	18.87	16.22	14.92	14.21
			$U(-\sqrt{3},\sqrt{3})$	ARL	135.19	119.48	98.06	63.47	34.78	22.57	16.32	13.02	11.2
				MDRL	71	69	64	44	25	14	8	5	3
				SDRL	157.69	129.02	97.48	62.48	33.32	23.60	18.44	15.61	13.99

, respectively. From these tables, it is seen that for $\tau =100$ and $\lambda =0.20$ and 0.25 (when $n=5$) and $\lambda =0.17$ (when $n=10$), the ARL₀ values are close to the corresponding zero-state values. However, this is not true for $\tau =500$ where the IC run-length characteristics are significantly smaller than the zero-state ones. Moreover, the OOC steady-state performance is very poor over the entire range of shifts. For example, when $n=5$ and $\lambda =0.25$, the zero-state ARL values of the DHWMA-SR chart under the $N(0,1)$ distribution are 370.45, 155.41, 70.30, 18.92, 6.36, 3.49, 2.35, 1.77 and 1.43 for $\delta =0,0.05,0.10,0.25,0.50,0.75,1.00,1.25$ and 1.50, respectively, while the corresponding steady-state ARL values for $\tau =500$ are 211.25, 129.77, 88.21, 56.22, 42.41, 34.75, 29.66, 26.70 and 25.05. Thus, although the steady-state ARL₀ value is very small, the ARL₁ values are too large.

6. Comparison Study

In this section, we compare the performance of the proposed chart with the control charts presented in Section 2 for $n=10$. It is to be noted that the different control charts are not compared under a fixed value of $\lambda$, but under an ARL₀ approximately equal to 370 and SDRL₀ ≤ ARL₀. The last constraint was proposed by Chan and Zhang [33], who suggested to take into account in the design of a control chart not only the ARL₀, but also the SDRL₀ in order to avoid a large variation in the run-length. Moreover, the proposed DHWMA-SR chart is compared to the nonparametric DHWMA sign and the parametric DHWMA-$\overline{X}$ charts under several symmetric and asymmetric distributions. The control chart with the smallest ARL₁ value in a specific shift is considered to be the most effective.

6.1. DHWMA-SR Versus EWMA-SR, DEWMA-SR and HWMA-SR

Table 7. Comparative study between DHWMA-SR, HWMA-SR, EWMA-SR and DEWMA-SR charts.

	Zero-State				Steady-State ($\mathit{\tau }=100$)				Steady-State ($\mathit{\tau }=500$)
	DHWMA-SR	HWMA-SR	EWMA-SR	DEWMA-SR	DHWMA-SR	HWMA-SR	EWMA-SR	DEWMA-SR	DHWMA-SR	HWMA-SR	EWMA-SR	DEWMA-SR
	$\mathit{\lambda }=\mathbf{0}.\mathbf{173}$	0.05	0.19	0.30	0.173	0.05	0.19	0.30	0.173	0.05	0.19	0.30
$\mathit{\delta }$	${\mathit{L}}_{\mathit{DH}}=\mathbf{1}.\mathbf{678}$	${\mathit{L}}_{\mathit{H}}=\mathbf{2}.\mathbf{308}$	${\mathit{L}}_{\mathit{D}}=\mathbf{2}.\mathbf{807}$	${\mathit{L}}_{\mathit{E}}=\mathbf{2}.\mathbf{681}$	${\mathit{L}}_{\mathit{DH}}=\mathbf{1}.\mathbf{678}$	${\mathit{L}}_{\mathit{H}}=\mathbf{2}.\mathbf{308}$	${\mathit{L}}_{\mathit{E}}=\mathbf{2}.\mathbf{807}$	${\mathit{L}}_{\mathit{D}}=\mathbf{2}.\mathbf{681}$	${\mathit{L}}_{\mathit{DH}}=\mathbf{1}.\mathbf{678}$	${\mathit{L}}_{\mathit{H}}=\mathbf{2}.\mathbf{308}$	${\mathit{L}}_{\mathit{E}}=\mathbf{2}.\mathbf{807}$	${\mathit{L}}_{\mathit{D}}=\mathbf{2}.\mathbf{681}$
0.00	370.50	370.14	370.67	370.66	369.80	345.11	367.54	366.50	172.74	202.37	370.52	371.43
	(369.34)	(322.32)	(368.89)	(370.37)	(344.82)	(303.07)	(367.90)	(365.20)	(222.37)	(221.42)	(371.27)	(371.65)
0.05	115.74	112.74	212.49	208.63	184.03	138.33	210.97	208.94	131.35	124.27	213.28	211.59
	(122.24)	(97.21)	(209.50)	(206.89)	(158.70)	(106.64)	(207.43)	(206.75)	(156.20)	(115.74)	(209.78)	(209.32)
0.10	43.91	44.22	88.07	85.47	107.47	70.55	88.66	86.37	97.99	78.93	88.54	86.66
	(44.06)	(35.24)	(84.10)	(82.13)	(83.14)	(47.06)	(84.36)	(82.37)	(107.06)	(64.44)	(84.23)	(83.00)
0.25	9.88	10.81	15.42	14.61	50.94	27.72	16.07	15.59	59.83	36.34	16.11	15.59
	(8.88)	(7.34)	(11.51)	(11.19)	(35.54)	(15.90)	(11.48)	(11.06)	(59.00)	(27.14)	(11.56)	(11.07)
0.50	3.21	4.09	4.76	4.47	29.58	13.85	5.53	5.73	38.70	17.48	5.53	5.73
	(2.54)	(2.06)	(2.39)	(2.36)	(19.94)	(7.60)	(2.48)	(2.22)	(37.49)	(13.75)	(2.48)	(2.21)
0.75	1.73	2.56	2.83	2.58	22.11	9.53	3.60	4.08	29.51	11.09	3.60	4.09
	(1.27)	(1.24)	(1.01)	(1.11)	(14.97)	(5.25)	(1.25)	(1.18)	(29.54)	(9.33)	(1.25)	(1.17)
1.00	1.21	1.83	2.22	1.88	18.67	7.62	2.91	3.50	24.69	8.30	2.91	3.50
	(0.64)	(0.98)	(0.48)	(0.71)	(12.76)	(4.25)	(0.88)	(0.91)	(25.53)	(7.40)	(0.87)	(0.91)
1.25	1.04	1.38	2.04	1.53	16.89	6.68	2.60	3.23	21.92	6.99	2.61	3.24
	(0.28)	(0.72)	(0.20)	(0.55)	(11.66)	(3.77)	(0.72)	(0.81)	(23.77)	(6.46)	(0.73)	(0.81)
1.50	1.01	1.14	2.00	1.31	15.97	6.19	2.46	3.11	20.57	6.31	2.46	3.11
	(0.11)	(0.45)	(0.06)	(0.47)	(11.09)	(3.52)	(0.67)	(0.76)	(22.13)	(5.99)	(0.67)	(0.76)

shows the ARL and SDRL values of the DHWMA-SR ($\lambda =0.173,L_{DH}=1.678$), HWMA-SR ($\lambda =0.05,L_H=2.308$), EWMA-SR ($\lambda =0.19,L_E=2.807$) and DEWMA-SR ($\lambda =0.30,L_D=2.681$) charts under the $N(0,1)$ distribution for both the zero- and steady-state ($\tau =100$ and 500) cases. The design parameters of the competing charts were chosen so as to satisfy the above conditions. We note that the SDRL₀ values of the EWMA-SR and DEWMA-SR charts with time-varying control limits are large for very small values of $\lambda$ and decrease as the value of $\lambda$ increases. The values of 0.19 (for the EWMA-SR chart) and 0.30 (for the DEWMA-SR chart) are the smallest, so that SDRL₀≤ ARL₀. From Table 7, we observe that for the zero-state scenario, the HWMA-SR chart is the most effective for a very small shift ($\delta =0.05$), while the DHWMA-SR chart is superior to its competitors for the rest of the range of shifts. It should be pointed out that both the HWMA-SR and DHWMA-SR charts have significantly better detection ability than the EWMA-SR and DEWMA-SR charts for small shifts ($\delta \le 0.25$). Moreover, the DEWMA-SR chart is more sensitive than the EWMA-SR chart over the entire range of shifts.

The results are different for the steady-state scenario. First of all, both of the EWMA-SR and DEWMA-SR charts have steady-state ARL₀ and SDRL₀ values approximately equal to the corresponding zero-state ones. This is not true for the HWMA-SR and DHWMA-SR charts when $\tau =500$. We note that the DHWMA-SR chart has ARL₀ $\approx 370$ when $\tau =100$, but its SDRL₀ value is smaller than the corresponding zero-state one (344.82 versus 369.34). Despite the fact that the DHWMA-SR and HWMA-SR charts have good OOC steady-state performance for $\delta =0.05$ and $\delta \le 0.10$, respectively, when $\tau =100$, they perform very poorly for larger shifts. Similar results are observed when $\tau =500$. Regarding the EWMA-SR and DEWMA-SR charts, it is seen that their steady-state performances are the same regardless of the value of $\tau$. Furthermore, the DEWMA-SR chart is more effective than the EWMA-SR chart for $\delta \le 0.25$ and vice versa for the rest of the range of shifts.

6.2. DHWMA-SR Versus DHWMA Sign and DHWMA-$\overline{X}$

Table 8. ARL values of the DHWMA-SR, DHWMA sign and DHWMA-$\overline{X}$ charts for $\lambda =0.15$ and $n=10$ under various symmetric distributions when $AR{L}_0\approx 370$.

$\mathit{\delta }$	Chart	${\mathit{L}}_{\mathit{DH}}$	Normal	${\mathit{t}}_4$	${\mathit{t}}_8$	Logistic	Laplace	CN	Uniform
0	DHWMA-SR	1.479	370.63	369.30	369.79	370.47	370.47	370.63	370.47
	DHWMA sign	1.504	372.69	368.86	371.16	370.89	370.89	369.90	370.89
	DHWMA-$\overline{X}$	1.551	370.51	295.79	347.46	353.25	335.31	414.71	386.46
0.05	DHWMA-SR	1.479	103.71	78.42	94.36	94.37	72.05	100.05	108.54
	DHWMA sign	1.504	212.55	93.78	116.86	115.81	66.19	129.55	190.18
	DHWMA-$\overline{X}$	1.551	104.61	103.08	104.99	105.28	104.71	107.83	105.94
0.10	DHWMA-SR	1.479	38.14	27.77	34.03	34.05	25.32	36.54	40.87
	DHWMA sign	1.504	52.12	33.66	44.10	43.43	23.78	50.21	84.02
	DHWMA-$\overline{X}$	1.551	38.33	38.64	37.94	38.27	38.41	38.70	38.55
0.25	DHWMA-SR	1.479	8.53	6.24	7.59	7.62	5.96	8.12	9.49
	DHWMA sign	1.504	11.90	7.55	9.83	9.82	5.98	11.46	20.50
	DHWMA-$\overline{X}$	1.551	8.38	8.27	8.19	8.29	8.30	8.25	8.31
0.50	DHWMA-SR	1.479	2.85	2.23	2.61	2.61	2.26	2.71	3.25
	DHWMA sign	1.504	3.93	2.70	3.40	3.31	2.47	3.74	6.32
	DHWMA-$\overline{X}$	1.551	2.70	2.65	2.71	2.66	2.67	2.65	2.69
0.75	DHWMA-SR	1.479	1.58	1.36	1.49	1.50	1.40	1.51	1.78
	DHWMA sign	1.504	2.17	1.59	1.92	1.90	1.58	2.05	3.20
	DHWMA-$\overline{X}$	1.551	1.46	1.42	1.46	1.45	1.45	1.43	1.46
1.00	DHWMA-SR	1.479	1.15	1.10	1.13	1.13	1.12	1.12	1.21
	DHWMA sign	1.504	1.46	1.20	1.35	1.34	1.23	1.39	1.98
	DHWMA-$\overline{X}$	1.551	1.09	1.09	1.09	1.09	1.09	1.08	1.09
1.25	DHWMA-SR	1.479	1.03	1.03	1.03	1.03	1.04	1.02	1.03
	DHWMA sign	1.504	1.16	1.06	1.11	1.11	1.08	1.12	1.33
	DHWMA-$\overline{X}$	1.551	1.01	1.02	1.01	1.01	1.01	1.01	1.01
1.50	DHWMA-SR	1.479	1.00	1.01	1.01	1.01	1.01	1.00	1.00
	DHWMA sign	1.504	1.04	1.02	1.03	1.03	1.02	1.03	1.03
	DHWMA-$\overline{X}$	1.551	1.00	1.00	1.00	1.00	1.00	1.00	1.00

shows the ARL values of the parametric DHWMA-$\overline{X}$ and nonparametric DHWMA sign charts for $\lambda =0.15$ and $n=10$ under several symmetric distributions. From this table, we observe the following:

• The nonparametric DHWMA charts are IC robust for all considered distributions, while the ARL₀ values of the DHWMA-$\overline{X}$ chart change when the underlying distribution also changes because it is designed for the normal distribution. For example, the ARL₀ of the DHWMA-$\overline{X}$ chart is 370.51 under the $N(0,1)$ distribution and ranges from 295 to 415 for the non-normal distributions.
• The DHWMA-SR chart outperforms the DHWMA sign chart for small to moderate shifts ($\delta \le 0.50$) and performs a slightly better for larger shifts under all considered distributions except for the Laplace distribution where, in this case, the DHWMA sign chart is more effective only for $\delta \le 0.10$.
• In the case of normal distribution, it is seen that the DHWMA-SR chart performs slightly better than the DHWMA-$\overline{X}$ chart for $\delta \le 0.10$ and vice versa for the rest of the range of shifts.
• The ARL₁ values of the DHWMA-$\overline{X}$ chart are approximately similar for all distributions. For example, when $\delta =0.10$, the ARL₁ values of the DHWMA-$\overline{X}$ chart are 38.33, 38.64, 37.94, 38.27, 38.41, 38.70 and 38.55 under the normal, $t_4$, $t_8$, logistic, Laplace, CN and uniform distributions, respectively.
• Although the ARL₀ values of the DHWMA-$\overline{X}$ chart are smaller than 370 under the Student’s t, logistic and Laplace distributions, the ARL₁ values of the DHWMA-SR chart are smaller than those of the DHWMA-$\overline{X}$ chart, especially for $\delta \le 0.50$, while their performance is similar for larger shifts.
• In the case of CN distribution where the ARL₀ of the DHWMA-$\overline{X}$ chart is equal to 414.71, the DHWMA-SR chart is more effective than the DHWMA-$\overline{X}$ chart for $\delta \le 0.25$, while the latter chart performs slightly better for larger shifts.
• In the case of uniform distribution where the ARL₀ of the DHWMA-$\overline{X}$ chart is equal to 386.46, the ARL₁ values of the DHWMA-$\overline{X}$ chart are smaller than those of the DHWMA-SR chart over the entire range of shifts.

In the following lines, we present the performance of the above competing control charts under several asymmetric distributions. Those are as follows:

• The gamma distribution, $Gam(\alpha ,\beta )$ with parameters $(\alpha ,\beta )=(1,1),(3,1)$ and $(5,1)$;
• The lognormal distribution $LN(\mu ,\sigma )$ with parameters $(\mu ,\sigma )=(0,0.25),(0,0.5)$ and $(0,1)$;
• The Weibull distribution $W(\alpha ,\beta )$ with parameters $(\alpha ,\beta )=(0.5,1),(1.5,1)$ and $(5,1)$.

All of these distributions, except for $W(5,1)$, are positively skewed. Furthermore, the distributions have been shifted and scaled, so that they have a median and standard deviation equal to 0 and 1, respectively.

Table 9. ARL values of the DHWMA-SR, DHWMA sign and DHWMA-$\overline{X}$ charts for $\lambda =0.15$ and $n=10$ under various asymmetric distributions when $AR{L}_0\approx 370$.

$\mathit{\delta }$	Chart	${\mathit{L}}_{\mathit{DH}}$	Gam(1,1)	Gam(3,1)	Gam(5,1)	LN(0,0.25)	LN(0,0.5)	LN(0,1)	W(0.5,1)	W(1.5,1)	W(5,1)
0	DHWMA-SR	1.479	19.64	48.91	71.93	97.52	39.12	16.77	8.66	43.94	217.71
	DHWMA sign	1.504	372.19	370.25	372.72	372.69	372.69	372.69	370.84	370.89	370.89
	DHWMA-$\overline{X}$	1.551	364.61	364.77	367.82	366.84	353.42	292.64	296.52	371.71	370.50
0.05	DHWMA-SR	1.479	10.82	22.33	29.11	35.18	17.98	7.23	2.88	21.10	222.38
	DHWMA sign	1.504	97.33	121.52	125.86	124.87	103.67	42.50	11.24	130.05	138.33
	DHWMA-$\overline{X}$	1.551	103.45	103.24	103.45	103.81	102.80	102.42	100.03	102.26	104.85
0.10	DHWMA-SR	1.479	6.88	12.84	15.74	17.99	10.39	3.99	1.05	12.43	61.27
	DHWMA sign	1.504	33.71	45.64	47.80	47.38	37.18	12.43	1.52	49.26	54.28
	DHWMA-$\overline{X}$	1.551	40.05	39.28	39.25	38.73	39.58	42.84	43.61	38.66	37.77
0.25	DHWMA-SR	1.479	2.68	4.50	5.20	5.59	3.68	1.24	1.00	4.53	10.68
	DHWMA sign	1.504	6.40	9.70	10.39	10.38	7.63	2.06	1.00	10.52	12.63
	DHWMA-$\overline{X}$	1.551	59.61	8.77	8.63	8.62	9.03	10.21	11.46	8.93	8.35
0.50	DHWMA-SR	1.479	1.08	1.73	1.98	2.08	1.42	1.00	1.00	1.76	3.29
	DHWMA sign	1.504	1.60	2.93	3.25	3.29	2.31	1.00	1.00	3.16	4.23
	DHWMA-$\overline{X}$	1.551	3.05	2.92	2.89	2.88	3.08	3.62	4.43	3.17	2.83
0.75	DHWMA-SR	1.479	1.00	1.09	1.17	1.22	1.02	1.00	1.00	1.09	1.77
	DHWMA sign	1.504	1.00	1.47	1.68	1.74	1.19	1.00	1.00	1.54	2.36
	DHWMA-$\overline{X}$	1.551	1.71	1.62	1.58	1.60	1.76	2.14	2.79	1.87	1.61
1.00	DHWMA-SR	1.479	1.00	1.00	1.01	1.02	1.00	1.00	1.00	1.00	1.23
	DHWMA sign	1.504	1.00	1.05	1.14	1.18	1.00	1.00	1.00	1.04	1.60
	DHWMA-$\overline{X}$	1.551	1.23	1.17	1.15	1.17	1.28	1.56	2.10	1.39	1.19
1.25	DHWMA-SR	1.479	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.05
	DHWMA sign	1.504	1.00	1.00	1.01	1.02	1.00	1.00	1.00	1.00	1.24
	DHWMA-$\overline{X}$	1.551	1.06	1.04	1.03	1.04	1.09	1.28	1.73	1.17	1.06
1.50	DHWMA-SR	1.479	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.01
	DHWMA sign	1.504	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.08
	DHWMA-$\overline{X}$	1.551	1.01	1.01	1.00	1.01	1.03	1.14	1.51	1.08	1.02

shows the ARL values of the DHWMA-SR, DHWMA sign and DHWMA-$\overline{X}$ charts under the above asymmetric distributions. It can be seen that the proposed DHWMA-SR chart is not IC robust because the assumption of symmetry is violated. On the other hand, as it was expected, the DHWMA sign chart is IC robust because the assumption of symmetry is not required. Moreover, we conclude that the ARL values of the DHWMA-$\overline{X}$ chart are comparable to those under the normal distribution when the underlying distribution approaches the normal. Finally, we note that the DHWMA sign chart is superior to the DHWMA-$\overline{X}$ chart for the $Gam(1,1),LN(0,0.5),LN(0,1)$ and $W(0.5,1)$ distributions.

7. Illustrative Examples

7.1. Smartphone Accelerometer Dataset

In this example, we use the smartphone accelerometer dataset, provided by Riaz et al. [35] and also applied by Tang et al. [12]. The accelerometer is widely used in many manufacturing and non-manufacturing processes to measure the vibrations in a car or airplane engine or in an earthquake. The dataset consists of 15 samples, each of size $n=10$, and is presented in

Table 10. Smartphone accelerometer dataset.

t	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$	${\mathit{x}}_9$	${\mathit{x}}_{10}$	${\mathit{SR}}_{\mathit{t}}$	${\mathit{E}}_{\mathit{t}}$	${\mathit{D}}_{\mathit{t}}$	${\mathit{H}}_{\mathit{t}}$	${\mathit{DH}}_{\mathit{t}}$
1	−6.6033	−8.6885	−5.8915	−5.3095	−8.4897	−6.5009	−5.7023	−7.0817	−7.7137	−6.2593	29	5.510	2.610	1.450	0.868
2	−7.2912	−7.0770	−6.6033	−6.7937	−5.5964	−6.6854	−7.0175	−8.1553	−7.9577	−7.7423	27	9.593	6.084	28.900	9.106
3	−7.1472	−4.8192	−7.9577	−6.9568	−6.9925	−5.8320	−7.7173	−4.2848	−8.1303	−7.1817	25	12.520	9.489	27.850	18.677
4	−6.2307	−7.3138	−6.5914	−9.2455	−8.1826	−8.6801	−6.8044	−4.8763	−7.8196	−7.2769	5	11.092	10.753	25.900	20.870
5	−8.1826	5.8701	−8.5778	−7.1448	−4.6407	−7.5007	−6.0010	−6.4176	−6.7901	−4.9751	33	15.254	13.375	22.075	21.502
6	−6.4545	−7.5507	−7.4162	−6.0986	−9.1538	−4.6109	−8.9110	−6.7925	−6.9401	−8.6302	5	13.306	13.906	22.860	21.391
7	−5.4226	−8.6837	−7.5947	−8.6432	−7.4781	−7.3578	−7.0317	−7.3531	−6.6366	−8.4278	−3	10.208	12.644	19.483	20.562
8	−7.3281	−8.3886	−5.7880	−7.1924	−7.9934	−7.6661	−6.9056	−8.3743	−8.9670	−7.1353	−9	6.558	10.078	15.971	19.329
9	−5.0822	−4.8846	−7.1758	−8.4540	−5.4048	−7.0163	−6.4581	−8.3100	−6.6187	−5.2988	35	11.962	11.064	15.050	19.352
10	−6.0010	−8.7111	−9.1169	−8.4957	−6.7568	−7.6554	−5.9820	−6.7592	−7.2603	−6.8532	5	10.639	11.001	15.767	18.597
11	−8.0029	−8.5016	−5.9796	−6.4640	−8.7111	−6.7878	−4.6990	−7.0175	−6.5759	−4.6728	25	13.368	12.230	15.690	18.611
12	−8.4314	−4.5347	−5.5547	−6.0641	−4.1598	−5.2453	−8.4350	−8.5826	−8.1826	−6.6068	29	16.338	14.342	16.736	18.702
13	−7.4959	−7.4400	−5.7975	−5.3500	−4.8989	−7.2162	−5.8011	−6.9865	−5.8904	−4.1277	49	22.544	18.496	18.758	19.423
14	−6.9270	−8.2433	−7.0175	−8.0220	−6.9639	−6.1308	−8.2957	−6.4259	−6.0034	−8.3897	11	20.350	19.856	19.185	18.886
15	−8.4040	−7.5173	−8.5457	−6.0379	−4.5264	−5.9844	−4.4597	−5.3976	−5.7523	−8.3040	35	23.134	21.886	19.800	19.462

. According to [12,35], the underlying distribution is symmetric and the IC process median is equal to −7.437. Thus, nonparametric control charts based on the signed-rank statistic can be applied. Setting an $AR{L}_0=370$ and SDRL₀ ≤ ARL₀, we construct the EWMA-SR ($\lambda =0.19,L_E=2.807$), the DEWMA-SR ($\lambda =0.30,L_D=2.681$), the HWMA-SR ($\lambda =0.05,L_H=2.308$) and the DHWMA-SR ($\lambda =0.173,L_{DH}=1.678$) charts. The charting statistics are also given in Table 10, while the control charts are shown in Figure 1, Figure 2, Figure 3 and Figure 4. The red dots indicate the charting statistics that lie over the $UC{L}_t$. We observe that the proposed chart detects the shift after three samples, the HWMA-SR chart after four samples and the EWMA-SR and DEWMA-SR charts after 13 samples. This example indicates the superiority of the proposed chart versus its competitors.

7.2. Piston Rings Dataset

In this example, we use the dataset from a piston ring process, provided by Montgomery [32]. At each sampling point t, 15 samples, each of size $n=5$, are collected in order to detect a shift in the mean/median of the inside diameter (in mm) of piston rings. The dataset is shown in

Table 11. Piston rings dataset.

t	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{SR}}_{\mathit{t}}$	${\mathit{E}}_{\mathit{t}}$	${\mathit{D}}_{\mathit{t}}$	${\mathit{H}}_{\mathit{t}}$	${\mathit{DH}}_{\mathit{t}}$
1	74.012	74.015	74.030	73.986	74.000	8	0.400	0.020	0.400	0.320
2	73.995	74.010	73.990	74.015	74.001	4	0.580	0.048	7.800	2.720
3	73.987	73.999	73.985	74.000	73.990	−14	−0.149	0.038	5.000	3.920
4	74.008	74.010	74.003	73.991	74.006	7	0.208	0.047	−0.283	3.053
5	74.003	74.000	74.001	73.986	73.997	−3	0.048	0.047	1.038	2.413
6	73.994	74.003	74.015	74.020	74.004	9	0.496	0.069	0.830	2.355
7	74.008	74.002	74.018	73.995	74.005	10	0.971	0.114	2.242	2.585
8	74.001	74.004	73.990	73.996	73.998	−6	0.622	0.140	2.550	2.258
9	74.015	74.000	74.016	74.025	74.000	12	1.191	0.192	2.381	2.665
10	74.030	74.005	74.000	74.016	74.012	14	1.832	0.274	3.550	3.063
11	74.001	73.990	73.995	74.010	74.024	4	1.940	0.358	4.095	3.052
12	74.015	74.020	74.024	74.005	74.019	15	2.593	0.469	4.636	3.584
13	74.035	74.010	74.012	74.015	74.026	15	3.213	0.606	5.500	3.954
14	74.017	74.013	74.036	74.025	74.026	15	3.803	0.766	6.231	4.311
15	74.010	74.005	74.029	74.000	74.020	14	4.313	0.944	6.807	4.613

. The IC process median ${\theta }_0$ computed from data known to be IC (see Table 6.3 of [19]) is equal to 74, while the Pearson’s coefficient of skewness is equal to 0.052, which is very close to 0. Thus, the distribution of the dataset is symmetric. Setting an $AR{L}_0=200$, we construct the EWMA-SR ($\lambda =0.05,L_E=2.267$), DEWMA-SR ($\lambda =0.05,L_D=1.726$), the HWMA-SR ($\lambda =0.05,L_H=1.924$) and the DHWMA-SR ($\lambda =0.20,L_{DH}=1.491$). These values of design parameters give the desired value of ARL₀. The charting statistics are also provided in Table 11 and the control charts are displayed in Figure 5, Figure 6, Figure 7 and Figure 8. From these figures, it is seen that the DEWMA-SR chart gives an OOC signal at the 13th sample and the other charts at the 12th sample. This example indicates that the proposed chart performs as well as or better than the other charts.

8. Conclusions and Recommendations

In this article, we proposed a new nonparametric DHWMA control chart based on the signed-rank statistic (DHWMA-SR) for monitoring shifts in the location parameter of an unknown but continuous and symmetric distribution. The run-length distribution and its associated characteristics were evaluated by performing Monte Carlo simulations for both the zero- and steady-state cases. It was shown that the performance of the proposed chart improves as the value of sample size n increases and as the value of smoothing parameter $\lambda$ decreases. However, the superiority of the DHWMA-SR chart for small values of $\lambda$ is accompanied with a large value of SDRL₀. For this reason, we recommend quality practitioners to design and implement the proposed chart with $\lambda$ = 0.20–0.25 for small values of n regardless of the amount of shift (which is usually unknown) and $\lambda \ge 0.15$ for n = 10–20 only for the zero-state case or if there is an evidence that the shift occurs very early in the process. Moreover, the performance of the DHWMA-SR chart was compared with that of the EWMA-SR, DEWMA-SR and HWMA-SR charts. The results indicated that the proposed chart has better detection ability than the EWMA-SR and DEWMA-SR charts, especially for small and moderate shifts, and it is superior to the HWMA-SR chart, especially for moderate and large shifts. We also compared the DHWMA-SR chart with the nonparametric DHWMA sign and the parametric DHWMA-$\overline{X}$ charts. The comparison study indicated that the proposed chart is more effective than the DHWMA sign chart under many symmetric distributions except for Laplace and, in many cases, it is more efficient than the DHWMA-$\overline{X}$ chart.

Knoth et al. [23] advised practitioners against the implementation of the HWMA chart and its extensions due to their poor steady-state performance. More specifically, they compared the HWMA and EWMA charts with design parameters that have arisen by setting as equal the IC asymptotic variances in their charting statistics. This criterion should be considered as arbitrary because the competing schemes have very different IC run-length characteristics. Alevizakos et al. [36] studied this topic and they concluded that an appropriately designed HWMA scheme is more effective than the EWMA chart for small and moderate shifts in the zero-state case. In this article, we showed that the steady-state IC run-length characteristics of the DHWMA-SR chart change as the shift occurs later and do not converge to specific values, like with the EWMA-SR and DEWMA-SR charts. Moreover, the steady-state OOC performance of the proposed chart, as well as that of the HWMA-SR chart, is very poor, especially for moderate and large shifts.

As future work, it would be of interest to investigate in more detail the steady-state performances of the HWMA and DHWMA schemes and develop modifications to enhance them. Additionally, the proposed scheme could be applied to joint monitoring of the unknown process parameters using the Lepage or Cucconi statistics.