George G. Zhyrnyy
GENERALIZED
NARROW LIMIT GAUGING METHOD
OF STATISTICAL QUALITY CONTROL
Presented
results are devoted to mathematical grounds of narrow limit gauging method
which is thought to be useful in attempts to reduce sample size in SQC by attributes.
Studying of narrow limit gauging method started about sixty years ago [1] and
was almost entirely devoted to case of single unknown parameter in different settings:
to use it with control charts [2], to design acceptance sampling plans [3],
etc. We shall develop the case of multiparameter
distribution of measured characteristic.
Problem setting. Assume that measured characteristic
is a random variable 
driven by law from
parametric family of distributions with probability density w.r.t.
Lebesgue measure on 
denoted as 
, where 
, multidimensional parameter 
is unknown, 
is a parametric set, 
is a connected subset
of 
. Assume also probability distribution function 
correspondent to to be continuous in 
. Assume desired quality level is 
- proportion of good
output units. Denote lower, resp., upper
specification as L, resp., U. So, null hypothesis is
that quality is good enough, mathematically, 
, where 
. Alternative hypothesis is 
, i.e. we produce too many nonconforming units.
Denote
narrow lower, resp., upper limit as 
, resp., 
, where 
. Actually, we are looking for such narrow limits that 
. Sample of size 
is taken and quantity
of pseudo-nonconforming units in a sample is counted. Assume 
and denote 
, 
, where 
.
If quantity of pseudo-nonconforming
units in a sample is greater than acceptan-
ce number 
then hypothesis 
should be accepted
else hypothesis 
should be accepted,
i.e. hypotheses about 
are tested instead of
hypotheses about 
. Denote 
, 
. If 
for some , 
then testing 
vs. 
can be reduced to testing
the 
vs. 
without any loss of
effectiveness. So is narrow limit gauging method. It has been studied in [4]. From
theorem 1 of [4] it is possible to derive that if is continuous and actually
depends on 2 or more components of 
then narrow limits gauging method cannot be constructed.
Now we consider the case 
.We shall call such method ‘generalized narrow limit gauging
method’ (GNLG method). Problem is to find numbers 
, 
as to make substitution
of pairs of hypotheses as reasonable as possible while , 
are given and
acceptance number is zero.
Main results. For the traditional testing
by attributes probabilities of errors of first and second types are
, 
.
Let us
consider GNLG method. Probability of error of first type is
=
Probability
of error of second type is
=
It is common fact that any machine,
production unit or device cannot operate perfectly and cannot ensure as small
variation as imaginable. So, there is least possible value for variation. It can
be shown that it is reasonable and possible to select parameters of GNLG as to
make 
and 
, thus having 
and 
. If both 
and then both 
and that is poor choice. The
benefit of giving up for growth of 
is significant reduction
of sample size. Practical rules to find the values of parameters of GNLG are
developed using numerical methods but table results are not given here due to
space limitations.
REFERENCES
1. Quality Control by Limit
Gauging // Production and Engineering
Bulletin. – v. 3, 1944. – P. 433–437.
2. Dudding B. P., Jennet W. J.
Control Chart Techniques when Manufacturing to a Specification. –
3. Beja A., Ladany
S. P. Efficient Sampling by Artificial Attributes //
Technometrics. – vol. 6, N 4, 1974. – P. 601–611.
4. Zhyrnyy G.G. Narrow limit
gauging method of SQC as a hypotheses testing problem// Applied Statistics.
Actuarial and Financial Mathematics. – 2003, ¹1-2. – Ñ. 87-95.