Asymptotic Equivalence in Nonparametric Function Problems-Theory and Applications

最近的研究结果表明，构造一个全面的渐近非参数函数估计理论是可能的。该建议的主要焦点是继续发展这样一个全面的渐近等价理论。特别强调的是建设性的、等价的结果。这使得研究人员能够在最容易处理的数学公式中开发统计方法，然后很容易地将这些方法转移到可能不太容易处理的实际兴趣公式中。此外，渐近等价的定义是一个非常严格的性质，在一些重要的实际情况下，特别是在涉及高维问题的情况下，它就失效了。该建议的另一个方面是发展一个较弱但仍具有建设性的部分等效理论，该理论可以在完全等效失效时有效地应用。还提出了利用等效思想进一步发展当前流行的样条和小波方法的研究。这将导致新的，改进的自适应平滑样条方法，并增加对现有的自适应小波方法的理解。计算和统计方法学的当代进展使在二维和三维数据环境中产生的复杂模式建模和分析成为可能。在医学、地球物理和天体物理成像以及与这些图像相对应的数据分析中，这样的例子比比皆是。与这些图像准确对应的统计模型是“非参数的”，因为它们不具有在前计算机时代发展的统计模型中普遍存在的限制性结构。目前的建议旨在对这些模型进行广泛而建设性的理解。更详细地说，目前有几种相当不同的方法来建模和分析这种非参数数据。本建议的主要重点是建立建设性地证明所有这些模型的操作相似性的结果。该建议的建设性性质应使从一种模式转向另一种模式成为可能。例如，这可以使选择一种最适合高效计算的形式，然后直接传递到另一种更有效的一般表示或进一步的数学分析。

Recent results indicate the possibility of constructing a comprehensive asymptotic theory of nonparametric function estimation. The major focus of the proposal is on continuing development of such a comprehensive asymptotic equivalence theory. In particular, the emphasis is on constructive, equivalence results. These allow researchers to develop statistical methodologies within the most tractable mathematical formulation and then to readily transfer those methodologies to the possibly less tractable formulation of practical interest. Additionally, asymptotic equivalence as so far defined is a very stringent property which fails in some important practical settings, particularly those involving higher dimensional problems. One additional aspect of the proposal is to develop a weaker, but still constructive, partial equivalencetheory which can productively be applied when full equivalence fails. Research involving the use of equivalence ideas to further develop currently popular spline and wavelet methodologies is also proposed. This should lead to new, improved adaptive smoothing spline methodologies and to increased understanding of the existing adaptive wavelet methodologies. Contemporary advances in computational and statistical methodology have made it possible to model and analyze complex patterns arising in two and three dimensional data contexts. Examples abound in medical, geophysical, and astrophysical imaging and in the analysis of the data corresponding to such images. The statistical models accurately corresponding to such images are "nonparametric" in the sense that they do not possess the restrictive structure prevalent in statistical models developed in the pre-computer era. The currentproposal is aimed at a broad, but constructive, understanding of such models. In more detail, there are currently several rather different ways to model and analyze such nonparametric data. The main focus of this proposal is on establishing results which constructively demonstrate the operational similarity of all these models. The constructive nature of the proposal should make it feasible to move from one model to the other. For example, this can make it possible to choose one form which is best for efficient computation and then to directly pass to another which is more efficient for general presentation or for further mathematical analysis.