Page 188 - Special Topic Session (STS) - Volume 3

STS539 Muhammad Abid et al.
∑ −1 ̅
̿ = =1 (6)
−1
−1
The value of ̅ equal to the target mean . The statistic given in (1) can also be
0
0
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rewritten as (cf. Abbas ):
1−
= ̅ + [( ) { ̅ + ̅ +. . . + ̅ }] (7)

−1 −1 −2 1
The statistic represented in (5) is to give weight to the current sample and to adopt
32
homogeneity in weight rest of all the samples are given weights 1 − (cf. Abbas ).
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Abbas defined the control limits of HWMA chart as follows:
2

− Κ√ , if = 1
2
0 0
=

2 (1−) 2
2
0
− Κ√ ( + ) if > 1
0 (−1)

= (8)
0
2 2
+ Κ√ , if = 1
0 0
=

2 2 (1−) 2
0
+ Κ√ ( + ) if > 1
0 (−1)
]
Based on the arcsine transformation the proposed non-parametric arcsine
HWMA (HWMANAS) chart is defined as:
= + (1 − ) ̅ −1 (9)

∑ −1
where ̅ = =1 and the value of = −1 (√). The statistic given in (9) can
−1
−1 0
also be rewritten as:
1−
= + [( ) { ̅ + ̅ +. . . + ̅ }] (10)

−1 −1 −2 1
The control limits of HWMANAS chart are defined as:
2

2
−1 (√) − Κ√ , if = 1
4 0
=

2

2
−1 (√) − Κ√ ( + (1−) 2 ) if > 1
0
(−1)

= −1 (√) (11)

2

2
−1 (√) + Κ√ , if = 1
0
= 4

2

2
0
−1 (√) + Κ√ ( + (1−) 2 ) if > 1
(−1) ]

where Κ controls the width of the control limits and it is selected based on the desired
in-control average run length.

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