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Differential privacy (DP) has become a rigorous central concept for privacyprotection in the past decade. We use Gaussian differential privacy (GDP) ingauging the level of privacy protection for releasing statistical summariesfrom data. The GDP is a natural and easy-to-interpret differential privacycriterion based on the statistical hypothesis testing framework. The Gaussianmechanism is a natural and fundamental mechanism that can be used to perturbmultivariate statistics to satisfy a $\mu$-GDP criterion, where $\mu>0$ standsfor the level of privacy protection. Requiring a certain level of differentialprivacy inevitably leads to a loss of statistical utility. We improve ordinaryGaussian mechanisms by developing rank-deficient James-Stein Gaussianmechanisms for releasing private multivariate statistics, and show that theproposed mechanisms have higher statistical utilities. Laplace mechanisms, themost commonly used mechanisms in the pure DP framework, are also investigatedunder the GDP criterion. We show that optimal calibration of multivariateLaplace mechanisms requires more information on the statistic than just theglobal sensitivity, and derive the minimal amount of Laplace perturbation forreleasing $\mu$-GDP contingency tables. Gaussian mechanisms are shown to havehigher statistical utilities than Laplace mechanisms, except for very lowlevels of privacy. The utility of proposed multivariate mechanisms is furtherdemonstrated using differentially private hypotheses tests on contingencytables. Bootstrap-based goodness-of-fit and homogeneity tests, utilizing theproposed rank-deficient James--Stein mechanisms, exhibit higher powers thannatural competitors.

Differentially Private Multivariate Statistics with an Application to Contingency Table Analysis