7. Probability and Statistics Soviet Essays - Sheynin, Oscar

article was considered either good enough or not. He then restricted his attention to the casein which the sample size was assigned and the batch accepted or not depending on whetherall the articles in the sample were good enough or at least one of them was not. As a result, anumber of accepted batches will then include defective articles. The main problem here wasto estimate the number of defective articles in the accepted, and in all the inspected batches.Kolmogorov provided an unbiased estimate of the accepted defective articles and concludedhis contribution by outlining how to apply his findings. In particular, he indicated theconsiderations for determining the sample size.Sirazhdinov [13; 17] methodologically followed Kolmogorov, but he considered a morecomplicated case in which a batch was rejected if the sample contained more than c rejectedarticles. Interesting here, in this version of the problem, is not only the estimate of anadvisable sample size, but also of an optimal, in some sense, choice of the number c. Theauthor offered reasoned recommendations for tackling both these questions.From among other contributions on acceptance inspection, we indicate the papers ofRomanovsky [112; 122], Eidelnant [12] and Bektaev & Eidelnant [1]. These authors werealso influenced by Kolmogorov.Mikhalevich [4; 5] studied sequential sampling plans. He also restricted his investigationby considering qualitative inspection when the acceptance/rejection of a batch depended onthe number of defective articles in the sample but the quantitative information on the extentof overstepping the limits of the technical tolerance or on other data important formanufacturing were not taken into account. Mikhalevich’s method of studying was based onWald’s idea of decision functions (1949).In our context, this idea is as follows. To be practical, the method of inspection should beoptimal in a number of directions which are to some extent contradictory. First of all, theinspection should ensure, with a sufficiently high reliability, the quality of the acceptedbatches. The cost of the inspection should be as low as possible. Then, the choice of the mosteconomical method of inspection should certainly take into account the peculiar features ofthe manufacturing and the nature of the inspected articles. It is therefore reasonable toassume as the initial data the cost of inspecting one article (c); the loss incurred whenaccepting a defective article (a); and the same, when a batch is rejected (B).Suppose that a batch has N articles, X of them defective. Then the mean loss incurred by itsacceptance isU X =ma(X – m)P[d 1 ; m|X] + BP[d 2 |X] + ckkP[ = k|X].Here, m is the number of recorded defective articles, P[d 1 ; m|X], the probability that the batchis accepted and m defective articles were revealed from among those inspected; P[d 2 |X], theprobability that the batch is rejected; and P[ = k|X] is the probability that the decision ismade after inspecting k articles. If the probability of X defective articles in a batch is (X),thenNu =X = 1U X (X)should be considered as the mean (unconditional) loss. The optimal method of inspection issuch for which this is minimal.Mikhalevich studied optimal methods of inspection assuming that the size of the batch waslarge and that consequently the hypergeometric distribution might be replaced by thebinomial law. Optimal here were certain repeated curtailed samples. A number of