FOR 1735:  Structural Inference in Statistics: Adaptation and Efficiency

• Subject Area: Mathematics
• Term: from 2012 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 197645397

Because of the increasing availability of data, statistics plays a more and more prominent role in many scientific and practical areas. On the other hand, e.g. due to high dimensionality or involved dependence structures, the most general realistic models are usually very complex so that poor statistical precision is very common in spite of the large amount of data. The Research Unit aims at developing new methods and tools to profit from structures underlying high-dimensional complex data. Structure is understood in a wide sense and not only concerns the unknown targets of inference (involving smoothness, sparsity or low dimensions), but also the data structure involving unknown correlation matrices, hierarchical or multi-scale interaction, dynamical systems or support properties of the noise. Typically, the precise form of the underlying structure is unknown. Hence, statistical methods should automatically adapt to the right structure among the class under consideration. That means that the procedures should exploit the underlying structure (nearly) as efficiently as if the family to which the true model component belongs were known. At the same time, the structures themselves often convey very important information, and their analysis can provide deeper insight into the nature of the data generating process. A typical example is given by assuming a low-dimensional structure of a "signal" embedded in a high-dimensional model for the random "noise": an estimation method needs to approximately discover the low-dimensional subspace, and then it yields an associated low-dimensional estimator. Similarly as in factor analysis, the subspace often hints at the essential forces driving the random mechanism. The long-term goal is to establish a general framework to adapt automatically and simultaneously to different structures, which might be present in the data. This will allow for much more efficient statistical procedures, which will have a strong impact on the significance of statistical results in a wide range of applications. The advanced new tools developed for the mathematical analysis will push on the whole field of mathematical statistics for complex structural models.

• DFG Programme: Research Units

Projects

• Central Project: Research Unit Structural Inference in Statistics: Adaption and Efficiency (Applicants Neumeyer, Natalie ;  Reiß, Markus )
• Efficient nonparametric regression when the support is bounded (Applicants Drees, Holger ;  Reiß, Markus )
• Multi-scale analysis of graphs (Applicants Blanchard, Gilles ;  von Luxburg, Ulrike )
• Multiple testing under unspecified dependency structure (Applicant Dickhaus, Thorsten )
• Semiparametric structural analysis in regression estimation (Applicants Neumeyer, Natalie ;  Spokoiny, Vladimir )
• Structural inference for high-dimensional covariance matrices (Applicants Dette, Holger ;  Rohde, Angelika )

• Spokespersons: Professor Dr. Holger Drees; Professor Dr. Markus Reiß