Bayesian Inference Estimation in Nonparametric Regression and its Frequentist Properties

非参数回归分析是当代一种迅速发展的统计方法。 它避免了经典回归方法中的一些限制性假设。 然而，到目前为止，它已经遭受了比较，从几乎完全缺乏的推理工具（例如测试和置信区间），这有助于使经典的回归分析如此有用。 Zhao（1998预印本）提出了非参数回归的第一个正确的贝叶斯估计，该估计对任何样本量都具有可接受的性能。 目前的建议是以各种方式扩展这种结构，并研究与这种贝叶斯公式相关的推理特性。 它的各种属性，包括其贝叶斯估计的可接受的性能，表明贝叶斯推断的基础上，这种先验分布及其扩展可能是可行的。 回归分析是一种经典的通用统计工具，它被广泛应用于统计应用的各个领域。 非参数回归分析是一种较新的方法，它避免了经典理论中的一些限制性假设。 一般理论包括回归、密度估计、信号处理和时间序列分析中的谱密度估计的非参数公式。该理论已经在医学和环境成像、经济分析、计算机工程和生物物理学等领域得到了重要的应用。 贝叶斯分析是一种非常强大的通用统计方法。 本研究计画提出一种使用贝氏技术解决非参数回归问题的可能方法。

Nonparametric regression analysis is a rapidly developing contemporary statistical methodology. It avoids some of the restrictive assumptions in more classical regression approaches. However, to date it has suffered by comparison from an almost total absence of the inferential tools (e.g. tests and confidence intervals) which help make classical regression analysis so useful. Zhao (1998 preprint) proposed the first proper Bayesian estimator for nonparametric regression which has acceptable performance for every sample size. The current proposal is to extend this construction in various ways and to study the inferential properties related to such Bayesian formulations. Its various properties, including the acceptable performance of its Bayes estimator, suggest that Bayesian inference based on this prior distribution and its extensions may be feasible. They may also present a satisfactory comprehensive set of inferential tools for nonparametric regression analysis and related formulations.Regression analysis is a classical general statistical tool, which has been widely used in virtually every area of statistical applications. Nonparametric regression analysis is a more recent methodology, which avoids some restrictive assumptions in the classical theory. The general theory includes nonparametric formulations of regression, of density estimation, of signal processing and of spectral density estimation in time series analysis. The theory has already found significant applications in areas such as medical and environmental imaging, economic analysis, computer engineering, and geophysics. Bayesian analysis is a very powerful general statistical methodology. This research project proposes a possible way of using Bayesian techniques to solve nonparametric regression problems.