Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The group testing estimation problem concerns estimating the number of defective elements $d$ in a collection of $n$ total within a fixed factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of $d$ within a constant factor requires $\Omega(\log n)$ queries. This confirms that a known $O(\log n)$ upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove a similar lower bound in the threshold query model.

翻译：高效计数或检测缺陷项是生物测试、质量控制、流算法等多个领域的关键任务。群组测试估计问题旨在以固定因子估计总数$n$中缺陷元素数量$d$。我们主要考虑经典查询模型，在该模型中，一次查询可揭示所选群组是否包含缺陷元素。我们证明，任何将$d$值估计在常数因子内的非自适应随机算法都至少需要$\Omega(\log n)$次查询。这证实了Bshouty（2019）已知的$O(\log n)$上界是紧的，并解决了Damaschke和Sheikh Muhammad（2010）的一个猜想。此外，我们在阈值查询模型中证明了类似的下界。