Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
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has the same mean and variance of the OC distribution for a given C kind plan. For the<br />
single <strong>sampling</strong> plan, let m 0 = n 0 p denote the parameter and let b = m 0 /m.<br />
Then<br />
E(m 0 ) = bE(m);<br />
V ar(m 0 ) = b 2 V (m).<br />
Equating the first two moments we get<br />
a 0 + 1 = E 2 (m)/V (m);<br />
n 0 = nE(m)/V (m).<br />
....(2.4.5)<br />
It may be noted here that, for a fixed ρ i ; i = 1, 2, ..., r the OC curve for a multiattribute<br />
C kind plan is looked upon as a function of np since in this case (p 1 , p 2 , ..., p r ) =<br />
(ρ 1 p, ρ 2 p, ..., ρ r p).<br />
In the following discussions we treat ρ i , i = 1, 2, ..., r as fixed.<br />
2.4.5 Sample size of the moment equivalent plan<br />
Hald (1981) assumes near identity of the OC curves of the moment equivalent single <strong>sampling</strong><br />
plan for estimating the probability of <strong>acceptance</strong> in case of a double <strong>sampling</strong> plan for<br />
different process average values. Hald’s assumption has been corroborated by numerical<br />
computations. The expression used by Hald may be considered as the first term of series<br />
expansion of Khamis (1960) and has the same accuracy as the Edgeworth and Cornish-<br />
Fisher approximation. Following Hald’s argument we may also assume the near identity of<br />
the OC curves i.e. the OC curve as of the given C kind plan as a function of p for a given<br />
vector (ρ 1 , ρ 2 , ..., ρ r ) and that of the OC curve of the moment equivalent single <strong>sampling</strong><br />
plan(SSP).<br />
We use the above logic (although mathematically non-rigorous) and numerically justify<br />
the near identity of the OC curves of a MASSP of C kind and that of the moment equivalent<br />
SSP (under the restriction of a specified set of values for the ρ i ’s). As we shall see later that<br />
we require effectively this near identity only for the tail probabilities.<br />
Theorem 2.4.2<br />
Let p [r] = (p 1 , p 2 , ..., p r ) and P (p [r] ) denote the OC function value at p [r] where there are r<br />
attributes.<br />
Further,<br />
P (p [r] ) = G(c 1 , np 1 )G(c 2 , np 2 )...G(c r , np r ) ≃ G(a 0 , n 0 (p 1 + p 2 + ... + p r ))<br />
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