Mathematical theory on statistical inference subject to randomization constraints

在许多现代估计问题中，收集数据的通信和处理受到严格的隐私条款的约束。因此，不是原始数据，而是在某种意义上修改或仅部分观察结果可用于统计推断。修改的程度-类似于假设检验的显著性水平-需要保持在规定值以上。在数学术语中，这种限制可以例如经由所谓的阿尔法局部差分隐私来表达。后者是以实际观测值为参数的操纵观测值的条件分布的变异性的度量。这种情况是显着不同的统计逆问题的设置，因为随机化机制是允许选择依赖于统计推断的意图。在这个项目中，我们将开发方法，以及获得深刻的理论理解的自适应推理问题下，这种随机约束的密度估计模型。这包括在多变量密度估计中的极大极小率，密度本身及其泛函的自适应估计的构造，以及置信带形式的自适应推断语句。

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.