  We consider the privacy problem of statistical estimation from distributed
data, where users communicate to a central processor over a Gaussian
multiple-access channel(MAC). To avoid the inevitable sacrifice of data utility
for privacy in digital transmission schemes, we devise an over-the-air
estimation strategy which utilizes the additive nature of MAC channel. Using
the mutual information between the channel outputs and users' data as the
metric, we obtain the privacy bounds for our scheme and validate that it can
guarantee strong privacy without incurring larger estimation error. Further, to
increase the robustness of our methods, we adjust our primary schemes by adding
Gaussian noises locally and derive the corresponding minimax mean squared error
under conditional mutual information constraints. Comparing the performance of
our methods to the digital ones, we show that the minimax error decreases by
$O(\frac{1}{n})$ in general, which suggests the advantages of over-the-air
estimation for preserving data privacy and utility.
