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 \emph{one} round of adaptivity is sufficient to achieve optimal mean-square error.
翻译:我们考虑在限制下估计对称对数计算分布的平均值的问题,因为从此分布中,每个样本只有一小点可供估测者使用。我们研究平均正方差,这是样本大小(因而也是位数)的函数。我们考虑三个设置:首先,一个集中的设置,一个编码器可以释放美元位数,给一个大小($)的样本,而对于这个设置,量化没有无惩罚性惩罚;第二,一个适应性设置,其中每个位数是当前观测和先前记录的位数的函数,我们在此设置中显示,与样本平均值相比,最佳相对效率恰恰是中位数的效率;最后,我们表明,在分布式设置中,每个位数只是局部样本的函数,没有一个估计器能够在参数空间上实现最佳效率。我们通过显示\emph{one}圆适应性足以达到最佳平均差错来补充我们适应性环境的结果。