Asymptotic Equivalence of Statistical Experiments

摘要题目：统计实验的渐近等价性实验是一组概率测度；可以在这些对象之间定义一个距离，如果这个距离很小，则它们包含的关于未知参数的信息是相似的。伪距离（或delta距离）这个基本缺陷是众所周知的；它推广了充分性的概念：如果实验通过充分性是等价的，那么它们的δ距离为零。数据简化的思想是充分统计概念的核心，它可以与概率极限定理相结合，从而通过高斯位置或泊松族近似一般统计模型。这些族允许在许多情况下显式地表示风险界限，然后这些风险界限在近似模型的渐近意义上变得有效。提出了将独立数据的非参数实验的近似范围从高斯扩展到无限可除实验，包括泊松情况。进一步，我们将研究一类关于渐近等价的随机过程实验。更多的见解也寻求与统计信息理论的联系，并从长远的角度来看，统计推断的其他主题似乎具有数学挑战性，如概率测量的微分几何和量子统计。因此，原理是用另一个更熟悉或更容易处理的统计模型来近似给定的统计模型。从最广泛的意义上讲，这与我们熟悉的通过正态分布或z分布（也称为贝尔曲线）近似简单随机样本的平均值有关，这在初级统计学课程中教授过，但是当我们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。两名博士生将持续参与该项目；其目的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。这个项目是锚定在数学系，有统计的主要研究传统。

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.