George G. Zhyrnyy

European University (Sevastopol branch), Ukraine

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. – London: British Standards Institution, 1955.

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.