Page 187 - Special Topic Session (STS) - Volume 3

STS539 Muhammad Abid et al.
14
Yang et al. observed that the in-control ARL results of EWMA NS chart is
not consistent when differs from 0.5. So, to overcome this deficiency, Yang
11
et al. also developed the non-parametric arcsine EWMA (EWMA NAS ) chart
+

using the arcsine transformation i.e., = −1 (√ ). The distribution of

1
follows the normal distribution with mean = −1 (√) and variance = ( )
4
(cf. Yang et al. ).
11
The plotting statistic of EWMA NAS chart is written as:
= + (1 − ) −1 (3)

Firstly, 0 = −1 (√) and the ( ) = −1 (√) and the

1 11
( ) = ( ), respectively (cf. Yang et al. ).
2− 4
So, the control limits of the EWMA NAS chart are:
1
= −1 (√) + √ ( ),
2− 4
= −1 (√) ,

1

( ).
= −1 (√) − √ 2− 4
2.3 Non-parametric sign CUSUM control chart
12
Using the statistic given in (1) Yang and Cheng developed the two
plotting statistic i.e., + and − of the non-parametric CUSUM sign
(CUSUMNS) chart as follows:
+
+
+
= (0, −1 + − ( + )) } (4)

0
+
−
−
= (0, −1 − ( − ) + )
0

where = 1, 2, . .. and initially, = 0 and = 0. The statistics given in (6)
+
−

are plotted against their control limits ℎ and −ℎ, respectively. The process is
considered to be out-of-control if ≥ ℎ or ≤ −ℎ, else, it is in-control.
−
+

2.4 Non-parametric mixed control chart
13
Abbasi et al. proposed a mixed nonparametric EWMA and CUSUM
(MECNAS) control charts using the arcsine transformation. The statistic of
MECNAS control chart is given below:
+
+
= max [0, ( − ) − + −1 } (9)

0

−
_
= max [0, −( − ) − + + −1

0

2.5 Proposed arcsine HWMA control chart
2
14
Hunter observed that the plotting statistic of EWMA chart proposed by Page
is given more weights to the current observations and fewer weights to the
32
previous observations. To overcome this deficiency, Abbas proposed a
HWMA chart and its statistic is defined as:
= ̅ + (1 − ) ̿ −1 (5)

where ̅ represent the sample mean for ℎ sample, ̿ −1 is the mean of the means of

previous − 1 samples and is the sensitivity parameter lies between zero and one
i.e., 0 < ≤ 1. The ̿ −1 is also defined as:
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