Mathematical theory on statistical inference subject to randomization constraints

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In many modern estimation problems, the communication and processing of collected data is subject to strict privacy provisions. Therefore, not the initial data but in a certain sense modified or only partial observations are available for the purpose of statistical inference. The degree of modification - similar to the significance level of a hypotheses test - is required to stay above a prescribed value. In mathematical terms, this restriction may be expressed, for instance, via the so-called alpha-local differential privacy. The latter is a measure for the variability of the conditional distribution of the manipulated observations given the actual observation value as a parameter. The situation is significantly different to the setting of statistical inverse problems, because the randomization mechanism is allowed to be chosen in dependence of the statistical inference intentions. In this project, we shall develop methodology as well as gain profound theoretical understanding of adaptive inference issues under such randomization constraints in the model of density estimation. This includes minimax rates in multivariate density estimation, the construction of adaptive estimators for the density itself and functionals thereof, as well as adaptive inference statements in form of confidence bands.