Estimating the number of defectives with group testing

Estimating the number of defective elements of a set has various biological applications including estimating the prevalence of a disease or disorder. Group testing has been shown to be more efficient than scrutinizing each element separately for defectiveness. In group testing, we query a subset of elements and the result of the query will be defective if the subset contains at least one defective element. We present an adaptive, randomized group-testing algorithm to estimate the number of defective elements with near-optimal number of queries. Our algorithm uses at most 2 log log d + O(1/δ
²
log 1/ε) queries and estimates the number of defective elements d up to a multiplicative factor of 1 ± δ, with error probability ≤ ε. Also, we show an information-theoretic lower bound (1 - ε) log log d - 1 on the necessary number of queries any adaptive algorithm makes to estimate the number of defective elements for constant δ.