In this brief note, we consider estimation of the bitwise combination $x_1 \lor \dots \lor x_n = \max_i x_i$ observing a set of noisy bits $\tilde x_i \in \{0, 1\}$ that represent the true, unobserved bits $x_i \in \{0, 1\}$ under randomized response. We demonstrate that various existing estimators for the extreme bit, including those based on computationally costly estimates of the sum of bits, can be reduced to a simple closed form computed in linear time (in $n$) and constant space, including in an online fashion as new $\tilde x_i$ are observed. In particular, we derive such an estimator and provide its variance using only elementary techniques.

翻译：在这篇简短笔记中，我们考虑在随机应答下通过一组噪声比特 $\tilde x_i \in \{0, 1\}$（表示未观测到的真实比特 $x_i \in \{0, 1\}$）来估计按位组合 $x_1 \lor \dots \lor x_n = \max_i x_i$ 的问题。我们证明，现有的各种极值比特估计方法（包括基于计算成本较高的比特和估计的方法）均可简化为一个简单的闭式解，该解可在线性时间（关于 $n$）和常数空间内计算，并支持以在线方式处理新观测到的 $\tilde x_i$。特别地，我们推导出这样的一个估计量，并仅使用初等技巧给出其方差。