Abstract

Three measuring devices are available to classify units into one of the two mutually exclusive categories. The first two devices are relatively inexpensive procedures which tend to classify sampling units incorrectly; the third device is, in general, an expensive procedure which classifies the units correctly. To estimate p, the proportion of units which belong to one of the two categories, a three phase sampling scheme is presented. At the first phase, a sample of n units is taken and fallible-2 classifications are obtained; at the second phase, a subsample of n l units is drawn from the first sample and fallible-l classifications are obtained; at the third phase, a subsample of n units is taken from Z the second sample and true classifications are obtained. Hypergeometric and multinomial variance and covariances are compared to justify that the observed frequencies to be denoted, n ijk, a jk, and x k can be assumed to be multinomially distributed. The maximum likelihood estimate of p and its asymptotic variance are derived. This variance is expressed in terms of the reliability coefficients of the fallible classifiers. The optimum values of n, nl ' and n which minimize the total 2 cost of selection and measurement for a fixed variance of estimation and which minimize the variance of the estimation for fixed budget are derived. This three phase sampling is compared both to a single phase sampling in which only true measurements are taken as well as to two phase sampling. Using a constructed finite population and various levels of reliabilities, we simulate the sampling and measurement operations to assess the correctness of our results. THREE PHASE SAMPLING FOR

Citations

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