Asymptotic Equivalence of Statistical Experiments

摘要DMS-0306497PI：Michael NussbaumTITLE：统计实验的渐近等价性实验是一族概率度量；可以定义这些对象之间的距离，以便如果该距离很小，它们包含的关于未知参数的信息也是相似的。这个基本的亏伪距离(或增量距离)是众所周知的；它推广了充分性的概念：如果实验通过充分性是等价的，则它们的增量距离为零。数据约简是充分统计概念的核心，它可以与概率极限定理相结合，从而得到用高斯位置或泊松族近似的一般统计模型。在许多情况下，这些族允许显式地表示风险界，然后这些风险界在近似模型中变得在渐近意义上有效。建议将独立数据的非参数试验的近似范围从高斯扩展到无穷可分试验，包括Poisson情形。此外，还将研究一类随机过程实验的渐近等价性。更多的洞察力也被寻求到与统计信息理论的联系，以及在更长期的角度，与统计推理中似乎在数学上具有挑战性的其他主题的联系，例如，概率度量的微分几何和量子统计。因此，原则是用更广为人知或更容易处理的另一种统计模型来近似给定的统计模型。在最广泛的意义上，这与人们熟悉的正态分布或Z分布(也称为贝尔曲线)对简单随机样本平均值的近似有关，这是在基础统计学课程中教授的，但当一个人必须处理非常高甚至无限维的数据或参数时，就会出现复杂的数学问题。两名博士生将不断参与该项目；其目标是培养具有明显数学兴趣的年轻统计学家，他们有望成为未来学术研究的领导者。这个项目的落脚点是一个数学系，该系在统计学方面有着重要的研究传统。

AbstractDMS-0306497PI: Michael NussbaumTITLE: Asymptotic Equivalence of Statistical ExperimentsAn 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.