THE
PERFO~~NCE
NO~~L
OF VARIABLE
S~1PLING
PLANS WHEN THE
DISTRIBUTION IS TRUNCATED
Helmut Schneider
University of North Carolina
Chapel Hill, N.C.
The robustness of standard variable sampling plans by Lieberman
and Resnikoff is considered with respect to a truncation of the
•
normal distribution.
It is shown how variable sampling plans
can be designed if the truncation point and
the unknown
0
0
are known.
For
case it is shown that the operating characteristic
curve is dependent upon the unknown
0.
INTRODUCTION
~lost
of the published sampling plans for inspection by variables,
e.g. the MIL-STD-4l4 and BS 6002, assume a normal distribution of the
inspected quality characteristic.
It is often the case that this assump-
tion is not justifiable, since there are some departures from the normal
distribution.
Some investigations concern the robustness of the variable
sampling plans if the skewness and the excess differ from normality.
A
summary of work on variable acceptance sampling with emphasis on nonnormality is given by Owen (1969).
Das and Mitra (1964) use the Cornish-
Fisher approximation up' to three terms to compute the corresponding probabilities, i.e. those of rejecting lots with an acceptable quality
level (AQL) and of accepting lots which have limiting quality (LQ).
A
non-normal distribution with known skewness and excess is assumed.
Masuda (1978), using simulation, investigates the robustness of normal
sampling plans applied to Student and Lognormal distributions.
Schneider
and Wilrich (1981) investigate the operating characteristic curve (OC) of a
variable sampling plan when the underlying distribution is the Betadistribution.
Furthermore it was pointed out that in the case of a Beta-
distribution the Cornish-Fisher approximation is rather poor, since the
probabilities of acceptance, computed by this method, are much too high.
Some papers deal with the design of variable sampling plans based
on specific distributions such as Gamma (Takagi (1972), Weibull
et ale (1980)).
(llosono
Srivastava (1961) considers the Cornish-Fisher approxi-
mation to design variable sampling plans.
2
In this paper the effect of truncation is investigated.
A
truncated normal distribution is suitable in many practical situations
where there is a restriction on the variable under consideration.
For
instance a truncated normal distribution arises in production engineering
when sorting procedures eliminate items above or below designated tolerance
limits.
Truncation often results from technical constraints of a produc-
tion process.
Firstly we will investigate the robustness of standard
variable sampling plans with respect to the truncation point.
Afterwards
we will deal with the design of variable sampling plans based on a truncated normal distribution with given variance
unknown
1.
0
0
2•
In the final section the
case is discussed.
Robustness of variable sampling plans
We assume the quality characteristic of the inspected item to be a
normally distributed random variable X with parameters
tion point x r , where
limit
U
is assigned.
0
2
and x r are known.
similarly.
U < x •
-
r
Items
Note that a lower limit L can be treated
=
x -~
o-lHx-~) /~ (_r_)
o
0
and
~(x)
x < x
-
r
(1)
o
~(x)
and trunca-
The p.d.f. of the quality characteristic X is given as
f(x)
where
2
A one-sided specification
Let us consider an upper limit
which have X > U are defective.
~, 0
x > x
r
are the standard normal p.d.f. and c.d.f. respectively.
The proportion of defective items in a lot is thus
..
3
x -v
If we let v
o
0
(U-v) 10 and
p =
=
p
•
x -v
H U- v ) ]/H_r_)
(~(_r_)
=
p
!::. =
1 - ~(v )/~(!::.+v
P
0
(x r -U)/o then we obtain
P
)
(2)
According to an agreement between the producer and the consumer, lots
with a fraction defective p
~
AQL are presumed to be good and ought to
be accepted with probability at least I-a.
Futhermore lots with p
>
LQ
are not acceptable to the consumer and should be rejected with probability
at least I-B.
Lieberman and Resnikoff (1955) established the following procedure
of variables acceptance sampling for the non-truncated case.
If the
specification is an upper limit, the value t
t
=
-x +
ko
(3)
of the test statistic T
U.
= X+
ko is compared with the specification limit
On the basis of this comparison, each lot is either accepted (t
or rejected (t
>
U).
~
U)
Accordingly, a variables plan is specified by the
parameters n (sample size) and k (acceptance constant).
The desired
(n,k)- plan has to fulfill the condition that the DC-curve of the plan
will pass through the points defined by (AQL, I-a) and (LQ,B).
To com-
pute the sample sixe n and the acceptance constant k, we have to analyse
the distribution of T for a given fraction
defective p.
The DC-curve
is given as
L(p)
=
peT ~ ulp)
(4)
4
If the variance of X is known, the statistic T is normally distributed.
Taking into consideration the requirements
L(AQL)
= peT
~ UIAQL)
L(LQ)
= peT
<
UILQ)
= l-cr
(5)
=e
(6)
it is not difficult to calculate the parameters (n,k).
There exist well
known published sampling plans such as MIL-STD4l4 and British STD6002.
In order to investigate the robustness of these sampling plans with
respect to a truncation, the DC-curve of these plans is examined for the
case where X is truncated at x.
Given a truncation point x
r
r
the expected
value and the variance of Tare
=
E[T]
-
~
aW(ur )
+
ka
( 7)
and
."
2
V[T]
a
n
=
[1 - W(u r ){W(ur )+ur }]
(8)
respectively, where (see Johnson and Kotz, 1970)
W(u r )
u
r
=
=
~(u )/~(u
r
(x -~)/a
r
=
r
)
(9)
~ +
vp
(0)
...
The DC-curve of an (n,k)-plan is then asymptotically given by
L(p)
where
=
~({v -k+W(~+v )}/ny(~+v
p
p
p
))
(11)
5
'Y(~+\I)
P
and
\l
p
r l-W(~+\I P){W(~+\I P)+1l+\IP}]-l
=
(12)
is the solution to equation (2).
Example
•
a
= 0.1
and 8
In the untruncated case we obtain the sampling plan n
= 34,
k
We consider the example AQL
= 0.01,
= 0.03,
LQ
= 0.1.
= 2.106.
Applying this plan to a truncated normal distribution the probabilities
of acceptance alter according to the amount of truncation.
Some proba-
bilities are presented in Table 1.
Table 1
e
..
II
L(AQL)
L(LQ)
< 0
0
0
0.05
0
0
001
0
0
0.2
0005
0
0 05
0.74
0.01
1.0
0.88
0.07
2.0
0.90
0.094
3.0
0.90
0.095
CXl
0.90
0.1
•
For small values of
~
the probabilities of acceptance differ signifi-
cantly from I-a and 8 respectively.
This implies for instance that if a
producer screens his production near the upper specification limit and
6
the consumer applies a standard variable sampling plan it is very likely
that the lot will be rejected.
In Figures 1 and 2 the OC-curves are plotted for two other examples
taken from BS 6002 which is very similar to the MIL-STD-414.
Curve I
shows the probability of acceptance of an (n, k) -plan if there is no
truncation, while curve II shows the probability of acceptance of the
same plan after a truncation.
2.
The design of a variable sampling plan for known a
We consider the statistic
T'
=
\l'"
(13)
kG
A
where
is the maximum likelihood estimate of
\l
The likelihood function
\1.
in the truncated case is
=
Since 0
2
n
exp(-I:
1
(x.-\l)
2
1
2
/20 )/[Hu
r
)12TI
0]
is assumed to be known the maximum likelihood estimate of
n
\l
(14)
is
the solution to
a In L =
a \l
a In ~ (u r )
-n - - : - - - - -
a
\l
n
+
I: (x.-\l)O
.1
1=
1
2
= o
(15)
'
This equation can be rewritten as
where W(x)
A]
7
[
X -lJ
...
lJ -
X - oW
= ~(x)/~(x).
=
.
•
0
(16)
Since (16) cannot be solved analytically, we
suggest using one of the following methods.
Figure 1. OC-curve
r--
(I) untruncated
of
plan(44,2.17)
Figure 2. OC-curve .of plan (8,1.96)
(II) truncated Li=O.2
(I) untruncated
o
~
-
0
C>
~
~
0
C>
0
C>
~
~
0
C>
·0
C>
r;
r;
0
Q)
Q)
s::
C>
ell
~
~
p.
0
ell
~
p.,
Q)
U
U
Q)
u
4-t
0
r-l
'ro!
.g
.c
a
~
~
0
<D
C>
<
0
tH
0
~
0
>.
.,.,
C>
0
~
>.
~
'r-i
0
u
s::
u
u
Li=O.2
C>
o
<
(II) truncated
.r-i
r-l
C>
•
.ro!
0
.c
0
0
.g
C>
~
--:
0
~
""!
0
0
~
0
-
C>
0
0
0
•
0.67
I.JJ
I
2.0{)
.
2.61
P,
JJJ
4.00
4.67
e
~~00
I
I
0.61
I
·1
I.JJ
iii
2.00·
~i
2.67
Pl.
•
i i i
J.B
4.00
i
I
4.61
8
Taylor Approximation
Let 6
=
(x -x)/o, then for 6
r
2 the Taylor approximation might be
>
used.
~
=
-x + oW(6)/(1+W'(6))
(17)
where
W' (6)
=
(18)
The error is then less than 7% of
'"
(~-x)/o
Rational Approximation
A
co~on
method of obtaining closed solutions for a nonlinear
equation with one variable, like (16), is to use rational functions to
approximate the functional relationship.
6
=
If we let 6
=
'"
(~-x)/o
and
'"
(x r -x)/o, then 0 is only a function of 8 and the estimates ~
.-
are
given by
'"
~
=
-x + 00 (6)
Several rational approximations of the form
o = Pn (8)/Pm(8)
+
€(8)
were tried, where P (8) and P (8) are polynoms of degree nand m respecn
m
tively, and £(8) is the approximation error.
The following approximation
for 0 is over the range of interest for 6, i.e. it is assumed that the
right truncation point is above the mean of the untruncated population.
•
9
For left truncation it is assumed that the left truncation point is below
the mean of the original population.
is truncated then 8
>
0.7979.
When less than 50% of the population
If 8 lies below 0.7979 the truncation is
so severe that a normal distribution should not be used since the variance
A
of
~
becomes too large.
If 8 is larger than 4.33 the truncation will be
ignored since the error is less than 10 -5 a.
For the range where the
Taylor approximation does not work we found the rational approximation
0.7979 < 8 <
The approximation error is 1~(8)1
<
1.3 10 -5 •
When
P 3 (8) = 17.79998379 - 19.306575258 + 7.229392418
=1
P (8)
Z
For the range
+
12.023480028 - 3.798744508
2 < 8 < 4.3
=1
2
- 0.935797428
3
2
the rational approximation
P3 (8) = 0.36123448 - 0.261369218
P2 (8)
2
+
0.063490238
- 0.927750538 + 0.417479268
5
was derived, where 1~(8)1 ~ 2.10- •
2
- 0.005171768
3
2
This approximation can be used as
an alternative to the Taylor approach, but the Taylor approximation
becomes more accurate as 8 increases.
Having an estimate of
•
~
the value
A
t'
=~
+
ka
is compared with the upper specification limit U.
comparison each lot is either accepted or rejected.
(19)
On the basis of this
In order to determine
10
a sampling plan such that its DC-curve passes through the points
(AQL,l-cr) and (LQ,B) the distribution of T' has to be evaluated.
It
A
is easily shown that
is asymptotically normal with mean
~
~
and variance
2
(J
Tl y(u r ), where y(u r ) is defined by
-1
y(u)
r
[1 - W(u r )(W(u r )+ur )]
=
(20)
and
u
r
=ll+v.
p
Hence the DC-curve is approximated by
peT' ~ VIp)
=
L(p)
= ~((v p -k)(n/y(ur ))~
(21)
and subsequently nand k are solutions to
= I-a
L(AQL)
= B
L(LQ)
Let u
l -cr
(22)
=
~
-1
(23)
(l-cr),
Us = ~ -1 (S)
and let v
AQL
' v
LQ
be the solutions
to equation (2) for p = AQL and p = LQ respectively, then we are able to
write the equations (22) and (23) in the form
1.
n'2
(v AQL-k)
y (ll+v AQL)
= u
~
l-cr
(24)
and
(vLQ-k)
and thus
n~
y(ll+V
LQ
)~
=
Us
(25)
•
11
n*
=
k*
=
where [x] is the integer part of x.
Tabe 2 shows the sampling plans for the case considered in Example 1.
TABLE 2
n*
k*
0.05
33
-0.79
0.1
18
0.67
0.2
19
1.47
0.5
27
1.96
1.0
32
2.08
2.0
34
2.10
m
34
2.106
~
e
It turns out that the appropriate sampling plan has an even lower sample
size than the standard sampling plan.
as some other examples show.
•
But this is not always the case
We have selected some sampling plans from
the British Standard Table III-A single sampling plans for normal inspection master table
'a' method. We have
c~lculated
the true probabilities
of acceptance for the case the normal distribution is truncated.
Further-
more we have evaluated formulas (26) and (27) to obtain the appropriate
12
sampling plans which are denoted as n* and k*.
Since the evaluation of these sampling plans are based on asymptotic
results we have performed a simulation study with 10,000 samples to calculate the probability of acceptance of the (n*,k*)-plans for small
sample sizes.
,
The results presented in Table 3 show that the suggested
plans are able to cope with the situation where a quality characteristic
is truncated normally distributed, given that the sample size n is large
enough to justify the use of the asymptotic results.
heavy, i.e. small
~
When truncation is
combined with large p values, the sample size has to
be larger than 20 in order to obtain accurate results by using the normal
approximation.
It can be seen from Table 3 that for small sample sizes
(code letter I) and truncation near the specification limit
the probability of accepting lots with p
= LQ
(~
= 0.2)
is significantly smaller
than assigned, while the probability of accepting lots with p
= AQL
is
~
met.
3.
The unknown a case
The maximum likelihood estimates of V and cr are solutions to the
equations
a In
av
L
=
0,
a In L
= o
a cr
(28)
The solution can be found by the Newt6n-Raphson method. To avoid iteration
we propose the following rational approximation.
Firstly, we shall re-
duce the two equations (28) which have to be solved simultaneously to
one equation.
T? do this we apply the following transformation due to
•
13
Table 3.
Simulation results
(n,k) plans taken from
BS 6002, L(Pl)=0.9, L(P )=0.1 for the untruncated case.
2
(n * ,k * ) derived by (26) and (27).
L(P ), L(P ), L* (PI) and L* (P ) are simulation results of
l
2
2
*
*
0.25
*
*
.20
• '5 0
1• 00
2.00
7 2.25 .0032 .0399 .593.0
. 853 .033
. 891 . 080
. 896 .093
5
6
7
7
1.73
2. 12
2. 2 3
2.2 LI
10~000
.893
.900
• 900
.900
samples.
.050
• 01:33
• 08 8
.096
*
*
*
I
*
***********************.*************~**********************.****
*
*
*
*
0.65
.20
.50
1• 00
2.00
8 1.96 .0080 .06L16
.476.0
.839 .028
. 891 .085
.898 .105
h
1.3L1
7 1.51
Po 1. 9 LI
t 9 1.98
.890
.893
• R9 q
.901
.040
.079
• 0 89
.085
*
*
*
*
I
*********.************.**.*****.****.***************************
*
.20
15 1.13 .0721 .212L1 .091.0
21
.05
.8R4
.066
*
*
.50
.744 .009
13
.79
.890
.076
*
* 1.00
.883 .060
14 1.05
.900
.Oe9
*
* 2.00
.899 .097
15 1.13
.904
.098
*********************************************************.******
*
.20
19 2."1 .003'5 .OI7!! .312.0
12 1.90
.890
.080
*
6 5
• 0
I
it
o. 25 *
•5 0
. 815 . 030
1 7 2. 3 0
• 89 7
• 0 89
* L
* 1.00
.886 .081
1 9 2.39
.902
.090
*
* 2. 00
. 894 .095 t 20 2. III
.905
• 09 1
*
**************.*************************************************
*
0.65
*
*
*
•2 0
.50
1.00
2.00
2 3 2. 1 2 • 0 0 8 6 • 0 324 . 139
.777
.883
.896
.0
.016
.074
.094
1 '3 1. 'I 8
19 1.97
22 2.09
232.11
• 900
.900
.912
.90b
• 08 9
.091
.098
.097
*
*
L
•
*
***********.***************************************.*********.**
*
.20
42 1.2!! .07'59 .1494.0
.0
46
.13
.891
.081
*
6.50
*
•
*
.50
1.00
2.00
.544
.859
_894
.002
.054
.095
34
.C)O
39 1.17
·~l.l3 1.24
.901
.e87
.901
.088
.095
.0t)3
*
*
*
L
********************************.*****.*.*********.*************
0.25
•
*
*
*
*
.20
.50
1.00
2.00
37 2."7 .0037 .0121
.115.0
.777 .019
.885 .077
.897 .094
22
31
36
37
].98
2.36
.891
2.1I5
.903
.698
2.'17
.90b
.01:3'1
.095
.099
.096
*
*
*
*
N
****************************************************************
0.65
*
*
"*
*
.20
.50
1.00
2.00
'I" 2.17
.0089 .0236 .025.0
.735 .012
.890 .083
.907 .108
25 1.57
37 2.04
'13 2.10
t/~5 2.18
.895
.902
.892
.902
.090
.090
.099
.09"
*
*
*
*
****************************************************************
.l..
I
Tois values deviate from the n values in the tables BS 6002 due to the method
of rounding the non integer values n * .
N
14
Cohen (1959).
Let
w
=
s
2
then the equations (28) can be reduced to an equation for u
w
=
r
1 - W(u r ) (W(ur)+ur )
[W(u )
r
..
(29)
- 2
(x -x)
r
+
(30)
u )2
r
Let
Q(w)
where u
r
=
is the unique solution to equation 30; then the maximum 1ike1i-
hood estimators can be written as
(31)
= x + Q(w) (xr -x)
(32)
The following rational approximation for Q(w) was derived which can be
used when the truncation is less than 50% of the population.
higher than 50% corresponds to w
(i)
>
Truncation
0.57081.
If w < 0.06246 (corresponds to u
r
> 4), then set Q(w)
= O.
The maximal absolute error is less than 10 -5 •
(ii)
If 0.06246 < w < 0.57081 then set
Q(w)
=
•
15
P4(w) = -0.00374615
2.87l68509w
0.17462558w
+
l7.48932655w 3 - ll.9171654w 4
.
P3 (w)
=1
+
2
+
5.74050l0lw - l3.53427037w
The maximal absolute error, is then !E(W)
I
2
3
+ 6.88665552w •
~ 5.10- 6 .
No approximation will be given for w > 0.57081, since the variances
of the estimators become extremely high.
Asymptotic Variance
The variance and covariance of the estimators ~" and 0"
are obtained
by inverting the Fisher information matrix
-E[a 2 ~nL/a 2~]
nJ
-E [a 2~nLj()~ao]
n
=
= 2
-E[a 2~nL/a~ao]
-E[a 2~nL/a 20]
o
where the elements J .. are given by
1)
(33)
=
-W(ur )[1
+
u r (W(ur )+ur )]
(35)
=
•
(34)
The asymptotic covariance matrix (ASeV) is thus
=
o
2
n
1
(36)
16
Variable Sampling Plans for Unknown
0
We consider a quality characteristic X which has a right truncated
p.d.f as given by (1), in which
specification limit
~ =
=1
But now, since
p
-
2 is assumed to be unknown. An upper
Let
is assigned.
r
v
p
=
(U-lJ) /0 and
(xr-U)/o; then we obtain as in Section 1
P
\l
U < x
0
and p.
H\l )
P
-
cI>(~+\l
P
(37)
)
is unknown, there is no one-to-one relationship between
0
We will proceed as in the known
,.
T
=
lJ
case and define the statistic
0
,.
+
ko
(38)
We know that T is asymptotically normally distributed with expected value
,.
E[T]
,.
=
E[lJ] + kE[o]
=
V[lJ] + k V[o] + 2k Cov[lJ,o]
(39)
and variance
V[T]
A
2
Ita
A
A
(40)
where V[lJ], V[o] and Cov[lJ,o] are given by (36).
In principle it is not difficult to find the distribution of T for
large n but the operating characteristic curve (OC) will unfortunately
depend on the unknown o.
.
The OC-curve is defined by
•
L(p)
=
peT ~ ulp).
Thus the OC-curve of an (n,k)-plan is approximately
( 41)
17
L(p)
= ~ ([n I
y (6+\.1 ,k)]
P
k
2
[(U-~)
(42)
IO-k])
where
( 43)
y(6+\.1 ,k) =
P
Since y(6+\.1 ,k) is not only a function of p but also of 6 and thus of
p
o the OC-curve is dependent on the unknown
0
2
•
Table 4
The Probability of Accepting a Lot with AQL and
LQ percent defectives for 0' = 0, .90, 1.10
e
k
L(LQ~
LQ
L(AQL!o')
.83
4.46
.900
.100
0.90
.839
.090
1 0 10
.934
.117
.900
.100
0.90
.877
.082
1.10
.913
.116
.900
.100
0.90
.893
.094
1.10
.905
.104
6
1.00
0.2
l.00
1.00
•
n
AQL
0'
0.5
1.0
20
1.48
26
1.95
30
2.05
.810
.865
3.90
3.97
A rough and ready method is to use the sample variance derived
from the lot history, in order to determine a sampling plan for given
risks (AQL,l-a) and (LQ,S).
Such an approach would be justified if L(p)
is not very sensitive to small changes of
are sufficiently large.
2
0 ,
given the sample sizes
A study of the OC-curve at AQL and LQ for var-
18
ious
~
and a shows that the DC-curve can change significantly when the stan-
dard deviation differs about 10% from the assumed standard deviation and
the trucation is heavy.
10%
For instance, when the true standard deviation
IS
lower than assumed, then the probability of accepting a lot at AQL
might decrease drastically as shown in Table 4.
then L(AQL) drops from .9 to .839.
For example, if
~
= 0.2
When the truncation is moderate the
differences might be ignored in applications.
In this paper we have considered only the singly truncated normal
distribution.
The performance of variable sampling plans for a doubly
truncated normal distribution can be investigated in a similar way. Unfortunately the estimating procedure becomes more complicated.
Only iter-
ative techniques are available.
Acknowledgement
The author's work was supported by the Deutsche Forschungsgemeinschaft
and the Air Force Office of Scientific Research and was completed while the
author was visiting the Statistics Department of the University of North
Carolina at Chapel Hill.
The author would like to thank the referees for
their valuable suggestions.
•
19
References
•
Chiu, W.K. and McP.Y. Leung
(1981) A graphical method for estimating
the parameters of a truncated normal distribution, Journal of
Quality Technology 13, 42-45.
Cohen, A.C. (1959) Simplified estimators for the normal distribution
when samples are singly censored or truncated, Technometrics 1,
217-237.
Cohen, A.C. (1949) On estimating the mean and standard deviation of
truncated normal distribution, JASA 44, 518-525.
Das, N.G. ann S.K. Mitra (1964) The effect of non-normality on sampling
inspection, Sankhya 261, 169-176.
Halperin, M. (1952) Maximum likelihood in truncated samples, Ann. Math.
Stat. 23, 226-238.
Hosono, Y., H. Okta, and S. Kase (1980) Design of single sampling plans
for doubly exponential characteristic, Proceedings of the First
Workshop on Quality Control, Berlin 1980.
Johnson, N.L. and S. Kotz (1970)
New York, Houghten Mifflin.
Continuous univariate distributions-I,
Lieberman, G.J. and G.J. Resnikoff (1955)
by variables, JASA 50, 457-516.
Sampling plans for inspection
Masuda, K. (1978) Effect of non-normality on sampling plans by Lieberman
and Resnikoff, ICQC, Tokyo, D3-7-D3-ll.
Owen, D.B. (1969) Summary of recent work on variables acceptance sampling
with emphasis on non-normality, Technometrics 11, 631-637.
Schneider, H. and P.Th. Wilrich (1981) The robustness of sampling plans
for inspection by variables, in Computational Statistics, H. Buning
and P. Naeve (eds.), Walter de Gruyter, Berlin, New York.
Takagi, K. (1972) On designing unknown-sigma sampling plans based on a
wide class of non-normal distributions, Technometrics 14, 669-678.
•
Srivastava, A.B.L. (1961) Variables sampling inspection for non-normal
samples, I.S.E.R., 145-152•
British Standard 6002, Sampling procedures and tables for inspection by
variables, 1980.
MIL-STD-4l4 Sampling procedures and tables for inspection by variables
for percent defectives MIL-STD-4l4. Washington: U.S. Government
Printing Office, 1957.