Page 186 - Special Topic Session (STS) - Volume 3

STS539 Muhammad Abid et al.
8-9
al. introduced the nonparametric EWMA and nonparametric CUSUM charts,
respectively, by employing the ranked set sampling (RSS) procedure.
The main disadvantage associated with the plotting statistic of an EWMA
chart is that it gives larger weights to the current observations and lesser
10
weights to the previous observations. To overcome this deficiency, Abbas
proposed a new chart named as homogeneously weighted moving average
(HWMA) chart. The plotting statistic of HWMA chart give a particular weight
to the current sample and the remaining weights are equally distributed
between the previous samples. To the best of our knowledge, there is no work
done in the literature on nonparametric HWMA chart. In this study, we develop
a nonparametric HWMA arcsine chart to fill this gap in the literature.

2. Methodology
2.1 Existing non-parametric control charts
In this section, structures of some existing non-parametric control charts
such as: EWMA NS , EWMA NAS , CUSUM and MEC NAS are presented.
NS
2.2 Non-parametric sign and arcsine EWMA control charts
Let , = 1, 2, … , be a randon sample of size obtained from a process

and has a process mean . Outline = − and then = ( >

0) ‘process proportion’. For in-control process we assume, = 0.5. The sign
statistic is defined as:
+
= ∑ , (1)

=1
where
1 if > 0

= { , = 1, 2, … , .

0 otherwise
Then the distribution of would follow binomial distribution with
+
parameters (, 0.5) for an in-control process. Therefore ( ) = 2 and
+
⁄
( ) = 4 . Based on (1) the statistic of non-parametric sign EWMA
+
⁄
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(EWMA NS ) chart proposed by Yang et al. is defined as:
+ = + (1 − ) + , (2)
+
−1
where 0 < ≤ 1 and indicates the value of in the sample. Initially
+
+
ℎ

the value of + is set equal to the mean of i.e., + = 2. The
+
⁄

0
0
mean and asymptotic variance of (2) are defined as: ( +) = 2 and
⁄

( +) = ( ).
2− 4
The control limits of the EWMA chart based on the asymptotic standard
NS
deviation of the statistic given in (2) are:

= + √ ( ), = , = − √ ( ).
+
2 2− 4 + 2 + 2 2− 4

where and are selected to obtain the desire in-control average run length
( ).

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