Asymptotic Equivalence of Nonparametric Experiments with Nuisance Parameters

渐进等价理论已成为数理统计研究的一个新兴领域。这个项目的目的是描述一些非参数曲线估计模型的渐近行为，这些模型的分布依赖于光滑函数f（x）。对于这些非参数模型中的每一个，研究者推导出一个极限模型，该极限模型可以用作渐近近似，使得在原始实验中的推断可以在极限实验下进行而不会丢失信息。许多规则的非参数模型已经被证明是近似的极限高斯白噪声漂移实验，观察布朗运动加上一个平均值，取决于f（x）。在这个项目中，研究人员考虑扩展的常规非参数问题，包括滋扰参数，如未知的方差或未知的样本点分布。在这些情况下的限制性实验包含描述滋扰参数的第二分量，这表明f（x）的估计可以与问题中的其他未知数分离。调查员还近似非参数问题的二维样本空间的布朗单过程加上一个平均值。这些渐近近似允许使用来自更简单的高斯模型的技术来解决非标准问题。在当今的许多技术领域，科学家很难分析不遵循典型模式的大量数据。这个问题出现在诸如解释医学图像和信号、描述植物物种分布以及分析金融序列等任务中。在这个项目中，研究人员将问题转化为可以通过现有统计工具解决的形式。因此，现有的分析方法可以应用于更广泛的技术问题。因此，研究者开发的技术可用于提高医学成像、信号解释、金融分析和其他应用的效率。

Asymptotic equivalence theory has emerged as a growing new area of mathematical statistics research. The purpose of this project is to describe the asymptotic behavior of some nonparametric curve estimation models, which have distributions that depend on a smooth function f(x). For each of these nonparametric models, the investigator derives a limiting model that can be used as an asymptotic approximation, such that inference in the original experiment can be performed under the limiting experiment without loss of information. Many regular nonparametric models have been shown to be approximable in the limit by a Gaussian white-noise-with-drift experiment which observes a Brownian motion plus a mean that depends on f(x). In this project, the investigator considers extensions of the regular nonparametric problem to include nuisance parameters such as an unknown variance or an unknown distribution of sample points. The limiting experiments in these cases contain a secondary component that describes the nuisance parameter, and this demonstrates that the estimation of f(x) can be separated from other unknowns in the problems. The investigator also approximates nonparametric problems with two-dimensional sample spaces by a Brownian sheet process plus a mean. These asymptotic approximations allow the non-standard problems to be solved using techniques from the simpler Gaussian models.In many areas of technology today, large sets of data that do not follow a typical pattern are difficult for scientists to analyze. This problem comes up in tasks such as interpreting medical images and signals, describing distributions of plant species, and analyzing financial series. In this project, the investigator transforms the problem into a form that can be solved by already existing statistical tools. As a result, existing methods of analysis can be applied to a wider range of technological problems. Thus the techniques developed by the investigator may be used to improve the efficiency of medical imaging, signal interpretation, financial analysis, and other applications.