Precision of double sampling estimators for comparing two probabilities

It is assumed that both error-prone and error-free observations can be made on a binary response, the second type being more expensive. In double sampling error-prone observations are made on all sample units and for a random subsample error-free observations are made as well. Tenenbein (1970) studied allocation of units in double sampling for estimating a single probability. We consider double sampling for comparing two probabilities. For different cost ratios for error-prone versus error-free observations and for fixed total cost, we assess how the large-sample variance of the log odds ratio estimator varies with the proportion of units subject to both error-prone and error-free observation. We determine the optimal double sampling proportion and the possible gain from using double sampling versus single sampling in which only error-free observations are made.