Mathematical theory on statistical inference subject to randomization constraints

在许多现代估计问题中，收集的数据的通信和处理都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后的或仅有部分观测值可用于统计推断。修改的程度--类似于假设检验的显著水平--被要求保持在规定的值之上。用数学术语来说，这种限制可以例如通过所谓的阿尔法-局部差异隐私来表示。后者是在给定实际观测值作为参数的情况下，对被操纵观测的条件分布的可变性的度量。这种情况与统计反问题的设置有很大的不同，因为允许根据统计推断意图选择随机化机制。在这个项目中，我们将对密度估计模型中随机化约束下的自适应推理问题进行方法论的发展和深入的理论理解。这包括多元密度估计中的极小极大率，密度本身及其泛函的自适应估计器的构造，以及以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.