An experiment is a family of probability measures; a distance can be defined between these objects such that the information they contain about an unknown parameter is similar if this distance is small. This basic deficiency pseudo-distance (or Delta-distance) is well known; it generalizes the concept of sufficiency: if experiments are equivalent via sufficiency then their Delta-distance is zero. The idea of data reduction which is at the heart of the concept of sufficient statistics can be combined with limit theorems of probability, resulting in an approximation of general statistical models by Gaussian location or Poisson families. These families allow explicit expressions for risk bounds in many instances, and these risk bounds then become valid in an asymptotic sense in the approximated models. It is proposed to extend the scope of approximations for nonparametric experiments of independent data from Gaussian to infinitely divisible experiments, including the Poisson case. Furthermore, a class of stochastic process experiments will be studied with regard to asymptotic equivalence. More insight is also sought into the connections to statistical information theory, and in a longer term perspective, to other topics in statistical inference which seem mathematically challenging such as e.g. differential geometry of probability measures and quantum statistics.
Thus the principle is to approximate a given statistical model by another which is better known or more tractable. In the widest sense, this is related to the familiar approximation of the mean of a simple random sample by a normal or Z-distribution (also known as Bell curve), taught in elementary statistics courses, but complex mathematical problems arise when one has to deal with data or parameters of very high or even infinite dimension. Two Ph. D. students will be constantly associated to the project; the aim is to educate young statisticians with distinctly mathematical interests who promise to be future leaders in academic research. This project is anchored in a mathematics department that has a major research tradition in statistics.
