Asymptotic Equivalence in Nonparametric Function Problems-Theory and Applications

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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.