Asymptotic Equivalence in Nonparametric Function Problems-Theory and Applications

最近的结果表明，建立一个全面的非参数函数估计的渐近理论的可能性。该提案的主要重点是继续发展这样一个全面的渐近等效理论。特别是，重点是建设性的，等价的结果。这些使研究人员能够在最易处理的数学公式中开发统计方法，然后将这些方法很容易地转移到可能不太容易处理的实际利益公式中。此外，迄今为止定义的渐近等价是一个非常严格的性质，在一些重要的实际设置中失败，特别是那些涉及高维问题。该建议的另一个方面是制定一个较弱的，但仍然是建设性的，部分等值理论，可以有效地适用于完全等值失败时。研究涉及使用等价的想法，以进一步发展目前流行的样条和小波方法也提出了建议。这将导致新的，改进的自适应平滑样条方法，并增加对现有自适应小波方法的理解。在计算和统计方法的当代进步已经使得有可能建模和分析在二维和三维数据上下文中出现的复杂模式。在医学、地球物理学和天体物理学成像以及对与这些图像相对应的数据的分析中，例子比比皆是。准确地对应于这些图像的统计模型是“非参数”的，在这个意义上，它们不具有在前计算机时代开发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式有一个广泛但建设性的理解。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。该建议的建设性应使从一种模式转向另一种模式成为可行。例如，这可以使得有可能选择一种形式，这是最好的有效计算，然后直接传递到另一个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.