Some limit results in estimation of proportion based on group testing

Group testing that tests groups with k experimental units instead of individuals has a long history and is very useful for estimating small proportion p under certain conditions. There are two sampling schemes for implementing group testing, one is Binomial sampling in which the number of groups n is predetermined, and another one is negative binomial sampling where the total number n of groups with a trait is predetermined. Many estimators including both the frequentist and Bayesian estimator have been proposed. The performance of all these estimators certainly depends on n and k. For the Bayesian estimators it also depends on the hyper-parameter $$\beta $$ β in the prior distribution $$Beta(1, \beta )$$ B e t a ( 1 , β ) of the proportion p. The present article studies the limits of all these estimators when n, or k, or $$\beta $$ β goes to infinity. The obtained results may be helpful with selecting n, k, and $$\beta $$ β .