Mean Estimation from Adaptive One-bit Measurements

TL;DR

This paper investigates the fundamental limits of estimating a normal distribution's mean using only one-bit measurements per sample, demonstrating that adaptive strategies require approximately 1.57 times more samples to achieve the same accuracy as unrestricted methods.

Contribution

It establishes the asymptotic lower bound on mean squared error for adaptive one-bit estimators and provides an explicit estimator that attains this bound.

Findings

Asymptotic mean squared error is at least /2n times the variance.

Adaptive one-bit estimation requires about /2 times more samples than unrestricted estimation.

An explicit estimator achieves the asymptotic optimal error bound.

Abstract

We consider the problem of estimating the mean of a normal distribution under the following constraint: the estimator can access only a single bit from each sample from this distribution. We study the squared error risk in this estimation as a function of the number of samples and one-bit measurements $n$. We consider an adaptive estimation setting where the single-bit sent at step $n$ is a function of both the new sample and the previous $n-1$ acquired bits. For this setting, we show that no estimator can attain asymptotic mean squared error smaller than $\pi/(2n)+O(n^{-2})$ times the variance. In other words, one-bit restriction increases the number of samples required for a prescribed accuracy of estimation by a factor of at least $\pi/2$ compared to the unrestricted case. In addition, we provide an explicit estimator that attains this asymptotic error, showing that, rather surprisingly, only $\pi/2$ times more samples are required in order to attain estimation performance equivalent to the unrestricted case.