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规定的约束。因此，不是初始数据，而是在一定意义上修改或只有部分观测值可用于统计推断。修正程度——类似于假设检验的显著性水平——需要保持在规定值之上。用数学术语来说，这种限制可以通过所谓的α -局部差分隐私来表达。后者是对给定实际观测值作为参数的操纵观测值的条件分布的可变性的度量。这种情况与统计逆问题的设置有很大的不同，因为随机化机制可以根据统计推理意图来选择。在本项目中，我们将开发方法，并对密度估计模型中这种随机化约束下的自适应推理问题获得深刻的理论理解。这包括多元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量的构造，以及以置信带形式的自适应推理陈述。

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.