. 2024 Mar 21;14:6759. doi: 10.1038/s41598-024-57407-1

Table 8.

The run-length characteristics (the first row contain ${ARL}_{1}$ s with ${SDRL}_{1}$ s in parenthesis, while ${MRL}_{1}$ s are in second row) of the existing MA-SN, EWMA-SN, MA-SR, EWMA-SR, MEC-SN, and proposed EWMA-MA(SR) control charts for $n = 10$ at ${ARL}_{0} \approx 370$ .

Control chart	$δ$	$AEQL$	$RMI$
	Normal distribution, i.e. $N (0, 1)$
MA-SN with $w = 5, k = 3.10$	173.1 (168.3) 122	57.3 (56.2) 40	23.5 (21.6) 17	12.0 (10.1) 9	7.3 (5.4) 6	3.4 (1.8) 3	2.3 (1.0) 2	1.5 (0.6) 2	1.2 (0.4) 1	1.0 (0.1) 1	10.8	0.91
MA-SR with $w = 5, k = 2.849$	140.9 (137.0) 100	39.6 (37.6) 28	15.7 (13.6) 11	8.2 (6.1) 6	5.3 (3.2) 4	2.9 (1.0) 3	2.2 (0.5) 2	2.0 (0.1) 2	2.0 (0.0) 2	2.0 (0.0) 2	14.3	0.75
EWMA-SN with $λ = 0.05, k = 2.675$	74.2 (59.7) 57	26.7 (15.3) 23	15.3 (7.0) 14	10.7 (4.2) 10	8.4 (2.8) 8	5.4 (1.5) 5	4.1 (1.0) 4	2.9 (0.6) 3	2.4 (0.5) 2	2.0 (0.1) 2	16.6	0.99
EWMA-SR with $λ = 0.05, k = 2.481$	56.1 (42.4) 44	20.8 (10.6) 18	12.5 (4.8) 11	9.1 (2.8) 9	7.2 (1.9) 7	5.1 (0.9) 5	4.3 (0.5) 4	3.9 (0.3) 4	3.5 (0.5) 3	3.0 (0.2) 3	21.6	1.14
MEC-SN with $λ = 0.05$ , $k = 0.5$ , $h = 51.28$	74.3 (38.5) 64	37.7 (11.6) 35	27.2 (6.0) 26	21.8 (3.9) 21	18.5 (2.9) 18	13.9 (1.8) 14	11.4 (1.3) 11	8.9 (0.8) 9	7.8 (0.6) 8	7.1 (0.3) 7	50.7	3.97
EWMA-MA (SR) with $L = 2.304,$ $w = 5, λ = 0.05$	50.4 (41.4) 40	17.4 (11.6) 15	9.6 (5.7) 9	6.3 (3.7) 6	4.5 (2.7) 4	2.5 (1.5) 2	1.6 (0.8) 1	1.1 (0.3) 1	1.0 (0.1) 1	1.0 (0.0) 1	7.8	0.0
Student’s t-distribution with $d f = 5$
MA-SN with $w = 5, k = 3.10$	131.0 (127.9) 93	37.2 (35.2) 27	14.9 (13.0) 11	7.8 (5.9) 6	5.0 (3.3) 4	2.7 (1.2) 3	2.0 (0.8) 2	1.4 (0.5) 1	1.2 (0.4) 1	1.1 (0.2) 1	9.6	0.67
MA-SR with $w = 5, k = 2.849$	113.7 (110.3) 80	28.5 (26.0) 20	11.3 (9.0) 9	6.3 (4.2) 5	4.3 (2.3) 4	2.6 (0.8) 2	2.2 (0.4) 2	2.0 (0.1) 2	2.0 (0.0) 2	2.0 (0.0) 2	13.6	0.71
EWMA-SN with $λ = 0.05, k = 2.675$	54.3 (40.2) 43	20.1 (10.3) 18	12.1 (4.9) 11	8.6 (3.0) 8	6.8 (2.1) 7	4.6 (1.2) 5	3.6 (0.9) 3	2.8 (0.6) 3	2.4 (0.5) 2	2.1 (0.3) 2	16.0	0.93
EWMA-SR with $λ = 0.05, k = 2.481$	45.9 (31.6) 37	17.6 (8.2) 16	10.8 (3.9) 10	7.9 (2.2) 8	6.5 (1.6) 6	4.8 (0.8) 5	4.2 (0.4) 4	3.8 (0.4) 4	3.5 (0.5) 4	3.2 (0.4) 3	21.8	1.22
MEC-SN with $λ = 0.05$ , $k = 0.5$ , $h = 51.28$	59.9 (26.8) 54	32.0 (8.2) 31	23.4 (4.5) 23	18.9 (3.0) 19	16.2 (2.3) 16	12.4 (1.5) 12	10.5 (1.1) 10	8.6 (0.7) 9	7.9 (0.6) 8	7.3 (0.5) 7	50.1	4.04
EWMA-MA (SR) with $L = 2.304, w = 5, λ = 0.05$	40.6 (32.2) 33	14.0 (8.8) 13	8.0 (4.7) 8	5.3 (3.1) 5	3.8 (2.3) 3	2.1 (1.3) 2	1.5 (0.8) 1	1.1 (0.3) 1	1.0 (0.2) 1	1.0 (0.1) 1	7.46	0.0
Laplace distribution, i.e. $L a p l a c e (0, \frac{1}{\sqrt{2}})$
MA-SN with $w = 5, k = 3.10$	81.4 (79.6) 57	21.0 (18.9) 15	9.4 (7.6) 7	5.5 (3.8) 4	4.0 (2.4) 3	2.5 (1.1) 2	2.0 (0.8) 2	1.5 (0.6) 1	1.3 (0.4) 1	1.1 (0.2) 1	9.1	0.40
MA-SR with $w = 5, k = 2.849$	83.1 (79.6) 59	21.1 (18.8) 15	9.0 (6.9) 7	5.4 (3.4) 4	3.9 (2.0) 3	2.6 (0.8) 2	2.2 (0.4) 2	2.0 (0.1) 2	2.0 (0.0) 2	2.0 (0.0) 2	13.3	0.61
EWMA-SN with $λ = 0.05, k = 2.675$	34.8 (22.2) 29	14.5 (6.4) 13	9.4 (3.4) 9	7.2 (2.3) 7	5.9 (1.7) 6	4.3 (1.1) 4	3.5 (0.8) 3	2.8 (0.6) 3	2.5 (0.5) 2	2.1 (0.3) 2	15.7	0.81
EWMA-SR with $λ = 0.05, k = 2.481$	37.4 (24.5) 31	15.3 (6.7) 14	9.8 (3.3) 9	7.5 (2.1) 7	6.2 (1.5) 6	4.7 (0.8) 5	4.2 (0.5) 4	3.8 (0.4) 4	3.6 (0.5) 4	3.2 (0.4) 3	21.8	1.21
MEC-SN with $λ = 0.05$ , $k = 0.5$ , $h = 51.28$	44.8 (16.0) 42	26.2 (5.6) 25	20.0 (3.4) 20	16.8 (2.4) 17	14.8 (2.0) 15	11.9 (1.4) 12	10.4 (1.1) 10	8.8 (0.8) 9	8.0 (0.6) 8	7.4 (0.5) 7	50.1	3.94
EWMA-MA (SR) with $L = 2.304, w = 5, λ = 0.05$	33.6 (25.9) 28	12.2 (7.6) 11	6.9 (4.1) 7	4.7 (2.9) 4	3.5 (2.2) 3	2.1 (1.2) 2	1.5 (0.8) 1	1.1 (0.4) 1	1.1 (0.2) 1	1.0 (0.0) 1	7.32	0.0
Logistic distribution, i.e. $L G (0, \frac{\sqrt{3}}{π})$
MA-SN with $w = 5, k = 3.10$	146.5 (144.0) 103	44.4 (42.6) 31	17.4 (15.5) 13	9.1 (7.2) 7	5.8 (4.1) 5	3.0 (1.4) 3	2.1 (0.8) 2	1.5 (0.6) 1	1.2 (0.4) 1	1.0 (0.2) 1	9.9	0.73
MA-SR with $w = 5, k = 2.849$	112.4 (110.7) 79	31.3 (28.7) 23	12.6 (10.3) 10	6.8 (4.7) 5	4.6 (2.7) 4	2.7 (0.9) 2	2.2 (0.5) 2	2.0 (0.1) 2	2.0 (0.0) 2	2.0 (0.0) 2	13.8	0.66
EWMA-SN with $λ = 0.05, k = 2.675$	61.5 (47.2) 48	22.2 (11.8) 20	13.2 (5.6) 12	9.3 (3.3) 9	7.4 (2.4) 7	4.9 (1.3) 5	3.8 (0.9) 4	2.8 (0.6) 3	2.4 (0.5) 2	2.1 (0.3) 2	16.3	0.93
EWMA-SR with $λ = 0.05, k = 2.481$	50.2 (36.1) 41	18.9 (9.1) 17	11.5 (4.2) 11	8.4 (2.5) 8	6.8 (1.7) 7	5.0 (0.9) 5	4.2 (0.5) 4	3.9 (0.4) 4	3.5 (0.5) 4	3.1 (0.3) 3	21.7	1.17
MEC-SN with $λ = 0.05$ , $k = 0.5$ , $h = 51.28$	65.8 (31.6) 58	34.0 (9.4) 32	24.6 (5.0) 24	20.0 (3.3) 20	17.0 (2.6) 17	13.0 (1.6) 13	10.9 (1.2) 11	8.8 (0.8) 9	7.9 (0.6) 8	7.2 (0.4) 7	50.4	3.96
EWMA-MA (SR) with $L = 2.304,$ $w = 5, λ = 0.05$	44.9 (36.2) 36	15.5 (10.1) 14	8.6 (5.0) 8	5.7 (3.4) 6	4.2 (2.5) 4	2.3 (1.4) 2	1.6 (0.8) 1	1.1 (0.3) 1	1.0 (0.1) 1	1.0 (0.0) 1	7.65	0.0
Contaminated Normal distribution with 10% contamination proportion
MA-SN with $w = 5, k = 3.10$	181.0 (177.4) 128	62.7 (60.5) 44	26.8 (24.7) 19	13.7 (11.7) 10	8.3 (6.6) 6	3.7 (2.1) 3	2.5 (1.1) 2	1.7 (0.6) 2	1.3 (0.5) 1	1.1 (0.3) 1	11.9	0.90
MA-SR with $w = 5, k = 2.849$	139.4 (134.0) 100	41.4 (39.2) 29	17.6 (15.2) 13	9.2 (7.0) 7	5.9 (3.7) 5	3.2 (1.3) 3	2.4 (0.6) 2	2.0 (0.2) 2	2.0 (0.0) 2	2.0 (0.0) 2	14.6	0.68
EWMA-SN with $λ = 0.05, k = 2.675$	80.0 (65.1) 61	28.9 (17.4) 25	16.4 (7.7) 15	11.6 (4.6) 11	8.9 (3.1) 8	5.7 (1.7) 6	4.3 (1.1) 4	3.1 (0.7) 3	2.6 (0.6) 3	2.1 (0.4) 2	17.7	0.94
EWMA-SR with $λ = 0.05, k = 2.481$	62.0 (46.6) 49	22.9 (12.0) 20	13.7 (5.6) 12	9.8 (3.2) 9	7.8 (2.2) 7	5.4 (1.1) 5	4.5 (0.6) 4	3.9 (0.3) 4	3.7 (0.5) 4	3.2 (0.4) 3	22.8	1.09
MEC-SN with $λ = 0.05$ , $k = 0.5$ , $h = 51.28$	78.0 (41.9) 67	39.4 (12.4) 37	28.2 (6.5) 27	22.7 (4.2) 22	19.3 (3.2) 19	14.5 (1.9) 14	11.9 (1.4) 12	9.3 (0.9) 9	8.2 (0.6) 8	7.4 (0.5) 7	53.0	3.79
EWMA-MA (SR) with $L = 2.304,$ $w = 5, λ = 0.05$	56.7 (48.5) 44	19.0 (12.7) 17	10.7 (6.5) 10	7.0 (4.1) 7	5.1 (3.0) 5	2.8 (1.7) 2	1.9 (1.0) 2	1.2 (0.4) 1	1.0 (0.2) 1	1.0 (0.1) 1	8.23	0.0

Control chart

δ

AEQL

RMI

0.10

0.20

0.30

0.40

0.50

0.75

1.00

1.50

2.00

3.0

Normal distribution, i.e.

N (0, 1)

MA-SN with

w = 5, k = 3.10

173.1 (168.3)

122

57.3 (56.2)

23.5 (21.6)

12.0 (10.1)

7.3 (5.4)

3.4 (1.8)

2.3 (1.0)

1.5 (0.6)

1.2 (0.4)

1.0 (0.1)

10.8

0.91

MA-SR with

w = 5, k = 2.849

140.9 (137.0)

100

39.6 (37.6)

15.7 (13.6)

8.2 (6.1)

5.3 (3.2)

2.9 (1.0)

2.2 (0.5)

2.0 (0.1)

2.0 (0.0)

14.3

0.75

EWMA-SN with

λ = 0.05, k = 2.675

74.2 (59.7)

26.7 (15.3)

15.3 (7.0)

10.7 (4.2)

8.4 (2.8)

5.4 (1.5)

4.1 (1.0)

2.9 (0.6)

2.4 (0.5)

2.0 (0.1)

16.6

0.99

EWMA-SR with

λ = 0.05, k = 2.481

56.1 (42.4)

20.8 (10.6)

12.5 (4.8)

9.1 (2.8)

7.2 (1.9)

5.1 (0.9)

4.3 (0.5)

3.9 (0.3)

3.5 (0.5)

3.0 (0.2)

21.6

1.14

MEC-SN with

λ = 0.05

k = 0.5

h = 51.28

74.3 (38.5)

37.7 (11.6)

27.2 (6.0)

21.8 (3.9)

18.5 (2.9)

13.9 (1.8)

11.4 (1.3)

8.9 (0.8)

7.8 (0.6)

7.1 (0.3)

50.7

3.97

EWMA-MA (SR) with

L = 2.304,

w = 5, λ = 0.05

50.4 (41.4)

17.4 (11.6)

9.6 (5.7)

6.3 (3.7)

4.5 (2.7)

2.5 (1.5)

1.6 (0.8)

1.1 (0.3)

1.0 (0.1)

1.0 (0.0)

7.8

0.0

Student’s t-distribution with

d f = 5

MA-SN with

w = 5, k = 3.10

131.0 (127.9)

37.2 (35.2)

14.9 (13.0)

7.8 (5.9)

5.0 (3.3)

2.7 (1.2)

2.0 (0.8)

1.4 (0.5)

1.2 (0.4)

1.1 (0.2)

9.6

0.67

MA-SR with

w = 5, k = 2.849

113.7 (110.3)

28.5 (26.0)

11.3 (9.0)

6.3 (4.2)

4.3 (2.3)

2.6 (0.8)

2.2 (0.4)

2.0 (0.1)

2.0 (0.0)

13.6

0.71

EWMA-SN with

λ = 0.05, k = 2.675

54.3 (40.2)

20.1 (10.3)

12.1 (4.9)

8.6 (3.0)

6.8 (2.1)

4.6 (1.2)

3.6 (0.9)

2.8 (0.6)

2.4 (0.5)

2.1 (0.3)

16.0

0.93

EWMA-SR with

λ = 0.05, k = 2.481

45.9 (31.6)

17.6 (8.2)

10.8 (3.9)

7.9 (2.2)

6.5 (1.6)

4.8 (0.8)

4.2 (0.4)

3.8 (0.4)

3.5 (0.5)

3.2 (0.4)

21.8

1.22

MEC-SN with

λ = 0.05

k = 0.5

h = 51.28

59.9 (26.8)

32.0 (8.2)

23.4 (4.5)

18.9 (3.0)

16.2 (2.3)

12.4 (1.5)

10.5 (1.1)

8.6 (0.7)

7.9 (0.6)

7.3 (0.5)

50.1

4.04

EWMA-MA (SR) with

L = 2.304, w = 5, λ = 0.05

40.6 (32.2)

14.0 (8.8)

8.0 (4.7)

5.3 (3.1)

3.8 (2.3)

2.1 (1.3)

1.5 (0.8)

1.1 (0.3)

1.0 (0.2)

1.0 (0.1)

7.46

0.0

Laplace distribution, i.e.

L a p l a c e (0, \frac{1}{\sqrt{2}})

MA-SN with

w = 5, k = 3.10

81.4 (79.6)

21.0 (18.9)

9.4 (7.6)

5.5 (3.8)

4.0 (2.4)

2.5 (1.1)

2.0 (0.8)

1.5 (0.6)

1.3 (0.4)

1.1 (0.2)

9.1

0.40

MA-SR with

w = 5, k = 2.849

83.1 (79.6)

21.1 (18.8)

9.0 (6.9)

5.4 (3.4)

3.9 (2.0)

2.6 (0.8)

2.2 (0.4)

2.0 (0.1)

2.0 (0.0)

13.3

0.61

EWMA-SN with

λ = 0.05, k = 2.675

34.8 (22.2)

14.5 (6.4)

9.4 (3.4)

7.2 (2.3)

5.9 (1.7)

4.3 (1.1)

3.5 (0.8)

2.8 (0.6)

2.5 (0.5)

2.1 (0.3)

15.7

0.81

EWMA-SR with

λ = 0.05, k = 2.481

37.4 (24.5)

15.3 (6.7)

9.8 (3.3)

7.5 (2.1)

6.2 (1.5)

4.7 (0.8)

4.2 (0.5)

3.8 (0.4)

3.6 (0.5)

3.2 (0.4)

21.8

1.21

MEC-SN with

λ = 0.05

k = 0.5

h = 51.28

44.8 (16.0)

26.2 (5.6)

20.0 (3.4)

16.8 (2.4)

14.8 (2.0)

11.9 (1.4)

10.4 (1.1)

8.8 (0.8)

8.0 (0.6)

7.4 (0.5)

50.1

3.94

EWMA-MA (SR) with

L = 2.304, w = 5, λ = 0.05

33.6 (25.9)

12.2 (7.6)

6.9 (4.1)

4.7 (2.9)

3.5 (2.2)

2.1 (1.2)

1.5 (0.8)

1.1 (0.4)

1.1 (0.2)

1.0 (0.0)

7.32

0.0

Logistic distribution, i.e.

L G (0, \frac{\sqrt{3}}{π})

MA-SN with

w = 5, k = 3.10

146.5 (144.0)

103

44.4 (42.6)

17.4 (15.5)

9.1 (7.2)

5.8 (4.1)

3.0 (1.4)

2.1 (0.8)

1.5 (0.6)

1.2 (0.4)

1.0 (0.2)

9.9

0.73

MA-SR with

w = 5, k = 2.849

112.4 (110.7)

31.3 (28.7)

12.6 (10.3)

6.8 (4.7)

4.6 (2.7)

2.7 (0.9)

2.2 (0.5)

2.0 (0.1)

2.0 (0.0)

13.8

0.66

EWMA-SN with

λ = 0.05, k = 2.675

61.5 (47.2)

22.2 (11.8)

13.2 (5.6)

9.3 (3.3)

7.4 (2.4)

4.9 (1.3)

3.8 (0.9)

2.8 (0.6)

2.4 (0.5)

2.1 (0.3)

16.3

0.93

EWMA-SR with

λ = 0.05, k = 2.481

50.2 (36.1)

18.9 (9.1)

11.5 (4.2)

8.4 (2.5)

6.8 (1.7)

5.0 (0.9)

4.2 (0.5)

3.9 (0.4)

3.5 (0.5)

3.1 (0.3)

21.7

1.17

MEC-SN with

λ = 0.05

k = 0.5

h = 51.28

65.8 (31.6)

34.0 (9.4)

24.6 (5.0)

20.0 (3.3)

17.0 (2.6)

13.0 (1.6)

10.9 (1.2)

8.8 (0.8)

7.9 (0.6)

7.2 (0.4)

50.4

3.96

EWMA-MA (SR) with

L = 2.304,

w = 5, λ = 0.05

44.9 (36.2)

15.5 (10.1)

8.6 (5.0)

5.7 (3.4)

4.2 (2.5)

2.3 (1.4)

1.6 (0.8)

1.1 (0.3)

1.0 (0.1)

1.0 (0.0)

7.65

0.0

Contaminated Normal distribution with 10% contamination proportion

MA-SN with

w = 5, k = 3.10

181.0 (177.4)

128

62.7 (60.5)

26.8 (24.7)

13.7 (11.7)

8.3 (6.6)

3.7 (2.1)

2.5 (1.1)

1.7 (0.6)

1.3 (0.5)

1.1 (0.3)

11.9

0.90

MA-SR with

w = 5, k = 2.849

139.4 (134.0)

100

41.4 (39.2)

17.6 (15.2)

9.2 (7.0)

5.9 (3.7)

3.2 (1.3)

2.4 (0.6)

2.0 (0.2)

2.0 (0.0)

14.6

0.68

EWMA-SN with

λ = 0.05, k = 2.675

80.0 (65.1)

28.9 (17.4)

16.4 (7.7)

11.6 (4.6)

8.9 (3.1)

5.7 (1.7)

4.3 (1.1)

3.1 (0.7)

2.6 (0.6)

2.1 (0.4)

17.7

0.94

EWMA-SR with

λ = 0.05, k = 2.481

62.0 (46.6)

22.9 (12.0)

13.7 (5.6)

9.8 (3.2)

7.8 (2.2)

5.4 (1.1)

4.5 (0.6)

3.9 (0.3)

3.7 (0.5)

3.2 (0.4)

22.8

1.09

MEC-SN with

λ = 0.05

k = 0.5

h = 51.28

78.0 (41.9)

39.4 (12.4)

28.2 (6.5)

22.7 (4.2)

19.3 (3.2)

14.5 (1.9)

11.9 (1.4)

9.3 (0.9)

8.2 (0.6)

7.4 (0.5)

53.0

3.79

EWMA-MA (SR) with

L = 2.304,

w = 5, λ = 0.05

56.7 (48.5)

19.0 (12.7)

10.7 (6.5)

7.0 (4.1)

5.1 (3.0)

2.8 (1.7)

1.9 (1.0)

1.2 (0.4)

1.0 (0.2)

1.0 (0.1)

8.23

0.0

Significant values are in bold.