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%24-differentially private projection-depth-based medians using the propose-test-release (PTR) and exponential mechanisms. Under general conditions on the input parameters and the population measure%2C (e.g. we do not assume any moment bounds)%2C we quantify the probability the test in PTR fails%2C as well as the cost of privacy via finite sample deviation bounds. Next%2C we show that when some observations are contaminated%2C the private projection-depth-based median does not break down%2C provided its input location and scale estimators do not break down. We demonstrate our main results on the canonical projection-depth-based median%2C as well as on projection-depth-based medians derived from trimmed estimators. In the Gaussian setting%2C we show that the resulting deviation bound matches the known lower bound for private Gaussian mean estimation. In the Cauchy setting%2C we show that the ``outlier error amplification'' effect resulting from the heavy tails outweighs the cost of privacy. This result is then verified via numerical simulations. Additionally%2C we present results on general PTR mechanisms and a uniform concentration result on the projected spacings of order statistics%2C which may be of general interest.?quality=80&w=800)
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