We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings: first, a centralized setting, where an encoder may release $n$ bits given a sample of size $n$ , and for which there is no asymptotic penalty for quantization; second, an adaptive setting in which each bit is a function of the current observation and previously recorded bits, where we show that the optimal relative efficiency compared to the sample mean is precisely the efficiency of the median; lastly, we show that in a distributed setting where each bit is only a function of a local sample, no estimator can achieve optimal efficiency uniformly over the parameter space. We additionally complement our results in the adaptive setting by showing that one round of adaptivity is sufficient to achieve optimal mean-square error.

中文翻译：

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一位测量的平均估计

我们考虑在以下约束条件下估计对称对数凹分布的平均值的问题：该分布中的每个样本只有一个比特可供估计器使用。我们研究均方误差作为样本大小（以及因此位数）的函数。我们考虑三个设置：第一，集中设置，编码器可以释放 $n$给定大小样本的位 $n$ ，并且量化没有渐近惩罚；第二，自适应设置，其中每个比特都是当前观察和先前记录的比特的函数，我们表明与样本均值相比的最佳相对效率恰好是中位数的效率；最后，我们表明，在每个比特只是局部样本的函数的分布式设置中，没有估计器可以在参数空间上均匀地实现最佳效率。我们还通过显示以下来补充我们在自适应设置中的结果一轮自适应足以实现最佳均方误差。