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OTE/SPH
OTE/SPH
Char Count= 0
3:7
August 31, 2006
JWBK119-21
CCC Scheme with Estimated Parameter 327
can be applied to the control limits; that is,
ln (α/2)
= γ φ , (21.5)
ln (1 − p 0 )
UCL γ φ
ln (1 − α/2)

= γ φ + 1, (21.6)
LCL γ φ
ln (1 − p 0 )
so that the ARL is maximized at p = p 0 , which is the in-control ARL, ARL 0 . It should
be noted that the above control limits do not have a direct probability interpretation;
that is, they do not always give the same probability content within the control limits.
Notably, eventhough thecontrol limits areshifted by a factor of γ φ , the type Irisk of the
chart is no longer given by the initial value φ. Moreover, the false alarm probabilities
beyond both sides of the control limits are no longer the same. The adjusted control
8
limits with equal probabilities are considered by Zhang et al. However, their proposed
procedure involves several iterations and only considers known p.
The actual type I risk α 0 can be recomputed by summing the tail probabilities

, where X is the number of conforming
α u = P X > UCL γ φ and α l = P X < LCL γ φ
items between two adjacent nonconforming items. From equations (21.5) and (21.6),
α u = (1 − p 0 ) ln(α/2)γ φ /ln(1−p 0 ) and α l = 1 − (1 − p 0 ) ln(1−α/2)γ φ /ln(1−p 0 ) ,

which can be expressed as
α u = (α/2) γ φ and α l = 1 − (1 − φ/2) , (21.7)
γ φ
respectively. It follows that

γ φ
γ φ
α 0 = α u + α l = (φ/2) − (1 − φ/2) + 1. (21.8)
As a result, α 0 is a nonlinear function of φ, and, despite the apparent relation between
φ and p 0 in equations (21.5) and (21.6), they are in fact independent of each other.

21.3 CCC SCHEME WITH ESTIMATED PARAMETER

In most circumstances, p 0 is not likely to be known and needs to be estimated. Tang
6
5
and Cheong presented a sequential estimation scheme, while Yang et al. considered
using the conventional binomial estimator of p for the CCC scheme.

21.3.1 Sequential estimation scheme
5
Tang and Cheong use the unbiased estimator of p given by Haldane, 9
m − 1
¯ p = , (21.9)
N m − 1
where the number of nonconforming items to be observed, m, is fixed a priori and the
total number of samples to be inspected, N m , is a random variable.