Asymptotically Sufficient Statistics in Nonparametric Curve Estimation

数理统计的一个重要领域是各种形式的非参数曲线估计问题。解决这些复杂问题的一个日益增长的方法是找到渐近充分的统计数据来简化它们。这些统计量也可以用来寻找渐近等价的实验，这些实验可以统一一些不同类型的非参数问题的推断和估计方法。一类非参数问题是回归模型，其中数据的平均值是观测数据的设计点的平滑函数。众所周知，这些模型可以被认为是连续的信号加白噪声模型的离散样本，其中，随着样本大小的增加，关于均值函数的信息丢失可以忽略不计。具体地说，白噪声模型观测到的连续过程的增量是渐近充分的，并且与回归观测具有几乎相同的分布。这个项目在一些非参数回归模型中找到渐近充分的统计量，这些模型允许估计讨厌的参数，例如误差的方差或协方差或随机放置的设计点的密度。这些统计数据由两部分组成：一部分对应于某个进程的增量，另一部分包含有关滋扰参数值的信息。这些渐近充分的统计导致一个渐近等价的模型，观察一个连续的高斯过程，其中的平均函数是第一次由观察到的过滤器转换。这些白噪声近似提供了一个新的和统一的方法，一些非参数回归问题。还有其他非参数模型，如独立数据的密度估计，已被证明是渐近等价于白噪声模型。找到滋扰参数的辅助估计量，然后近似条件分布给出这些估计量的相同的技术也可以应用于密度估计实验。这些充分的统计量是在寻找近似的方向上的一个步骤的密度估计实验上的二维sample spaces.This项目的目的是找到更好的技术，估计信号中存在的噪声，通过构建近似，是适合于大样本大小。这项工作将对非参数曲线估计理论产生影响，这是分析金融序列或医学图像等应用的基础，也将对数理统计的研究和教育产生广泛的影响。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。像这样的非规则估计问题可能会出现在诸如过滤噪声医学图像或信号、估计植物物种分布或分析金融序列等问题中。信号解释、金融分析和其他应用。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。

An important area of mathematical statistics are nonparametric curve estimation problems in their various forms. One growing approach to these complicated problems is to find asymptotically sufficient statistics to simplify them. These statistics can also be used to find asymptotically equivalent experiments that can unify the approaches to inference andestimation for a number of different types of nonparametric problems. One class of nonparametric problems is the class of regression models where the mean of the data is a smooth function of the design points at which the data is observed. It is well known that these models can be thought of as discrete samples of a continuous, signal-plus-white-noise model where, as the size of the sample increases, negligible information is lost about the mean function. Specifically, the increments of the continuous process observed by thewhite-noise model are asymptotically sufficient and have nearly the same distribution as the regression observations. This project finds asymptotically sufficient statistics in some nonparametric regression models that allow estimation of nuisance parameters such as the variance or covariance of the errors or the density of the randomly placed design points. These statistics consist of two parts: one corresponds to the increments of some process, and the second contains information about the value of the nuisance parameter. These asymptotically sufficient statistics lead to an asymptotically equivalent model that observes a continuous Gaussian process where the mean function is first transformed by an observed filter. These white-noise approximations provide a new and unifying approach to a number of nonparametric regression problems. There are other nonparametric models, such as density estimation from independent data, that have been shown to be asymptotically equivalent to the white-noise model. The same technique of finding auxiliary estimators of nuisance parameters and then approximating the conditional distribution given these estimators can also be applied to the density estimation experiment. These sufficient statistics are a step in the direction of finding an approximation to the density estimation experiment on two dimensional sample spaces.This project seeks to find better techniques for estimating a signal in the presence of noise by constructing approximations that are appropriate for large sample sizes. The work will have an impact on the theory of nonparametric curve estimation, the basis for applications such asanalyzing financial series or medical images, as well as a broad impact on research and education in mathematical statistics. In many areas of technology today, large sets of data that do not follow a typical pattern are difficult for scientists to analyze. Non-regular estimation problems like this might come up in problems such as filtering noisy medical images or signals, estimating distributions of plant species, or analyzing financial series. signal interpretation, financial analysis, and other applications. 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.