Asymptotic Equivalence of Nonparametric Experiments with Nuisance Parameters

渐近等价理论是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一个光滑函数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，该模型可以用作渐近逼近，这样原始实验中的推理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一个高斯白噪声漂移实验证明在极限上是近似的，该实验观察到布朗运动加上依赖于f(x)的平均值。在这个项目中，研究者考虑了正则非参数问题的扩展，以包括诸如未知方差或样本点的未知分布等令人讨厌的参数。在这些情况下，极限实验包含一个描述干扰参数的次要分量，这表明f(x)的估计可以与问题中的其他未知数分开。研究者还近似非参数问题与二维样本空间的布朗页过程加一个平均值。这些渐近逼近允许使用更简单的高斯模型的技术来解决非标准问题。在当今许多技术领域，不遵循典型模式的大量数据对科学家来说很难分析。在解释医学图像和信号、描述植物物种分布和分析金融序列等任务中都会出现这个问题。在这个项目中，研究者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、财务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.