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In some situations, it is desired to estimate multinomial proportions from data which have been misclassified. One such area is the sampling inspection area of quality control. In this paper, it is assumed that two measuring devices are available to classify units into one of r mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. In order to estimate the proportions pi (i = 1,2, …, r) a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimates of the pi are derived along with their asymptotic variances. Optimum values of n and N which minimize the measurement costs for a fixed precision of estimation and which minimize the precision for fixed cost are derived. The procedures of estimation and allocation of sample size are illustrated by an example in quality control.
Aaron Tenenbein (Tue,) studied this question.