  In this paper, we investigate the differentially private estimation of data
depth functions and their associated medians. We introduce several methods for
privatizing depth values at a fixed point, and show that for some depth
functions, when the depth is computed at an out of sample point, privacy can be
gained for free when $n\rightarrow \infty$. We also present a method for
privately estimating the vector of sample point depth values. Additionally, we
introduce estimation methods for depth-based medians for both depth functions
with low global sensitivity and depth functions with only highly probable, low
local sensitivity. We provide a general result (Lemma 1) which can be used to
prove consistency of an estimator produced by the exponential mechanism,
provided the limiting cost function is sufficiently smooth at a unique
minimizer. We also introduce a general algorithm to privately estimate a
minimizer of a cost function which has, with high probability, low local
sensitivity. This algorithm combines the propose-test-release algorithm with
the exponential mechanism. An application of this algorithm to generate
consistent estimates of the projection depth-based median is presented. Thus,
for these private depth-based medians, we show that it is possible for privacy
to be obtained for free when $n\rightarrow \infty$.
