Optimal sequential plans based on prior distributions and costs
A. Hald in Technometris, Vol. 2, No. 3 proposed a model of sampling inspection by attributes. He postulated certain definite forms for the losses which are associated with the acceptance or rejection of a lot, and for the cost of taking a single observation. He further assumed that there exists an a priori distribution of the number of defectives, the distribution being known to the experimenter. Under these assumptions he derived the associated single sampling plans which minimize the expected costs averaged with respect to the known a priori distribution. J. Pfanzagl in Technometrics, Vol. 5, No. 2 specialized Hald's model by considering a particular type of a priori distribution, and obtained the double sampling plans which are optimal in the same sense. In this paper, Pfanzagl's work has been generalized by extending his results to sequential plans. In particular, the related optimal (Bayes) sequential sampling plans and their operating characteristics are derived. The Bayes risks of the optimal sequential plans are compared with those of the corresponding optimal single sampling plans for certain representative values of the parameters. The limiting behavior for large lots of the optimal boundaries for the sequential sampling plans is studied using a particular normalization due to Chernoff and Ray. (Annals of Math. Stat., Vol. 36, No. 5). (Author).
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accept before taking Acceptance and Rejection apriori distribution ASN-functions for selected Bayes procedure Bayes risk Bayes sequential procedure Bayes single sampling binomial Chernoff and Ray compute conjugate priors consumer's risk continuation region corresponding Costs due decision problem defective item double sampling plans exact stage fraction defective free boundary problem given Hald Hald's model hypergeometric distribution Largest number lattice points Lemma loss function neutral line Normal distribution number of defectives number of observations observations the plan obtain optimal boundary optimal sequential plan optimal single optimum P L A N P)-plane parameters partial differential equation Pfanzagl's Model plan cannot accept plan cannot reject point of truncation Polya distribution posteriori distribution random variables random walk reject before taking Rejection numbers sampling inspection plans selected values sequential analysis sequential sampling plans single sampling plan stage of truncation stopping point stopping region Theorem upper bound upper boundary Wiener Process