Adaptively local I - dimensional subproblems

We provide a new insight of the difficulty of nonparametric estimation of a whole function. A new method is invented for finding a minimax lower bound of globally estimating a function. The idea is to adjust automatically the direction to the nearly hardest I-dimensional subproblem at each location, and to use locally the difficulty of I-dimensional subproblem. In a variety of contexts, our method can give not only attainable global rates, but also constant factors. Comparing with the existing techniques, our method has the advantages of being easily implemented and understood, and can give constant factors as well. We illustrate the lower bound by using examples of nonparametric density estimation as well as nonparametric regression. Concise proofs of the lower rates are given. Applying our lower bound to deconvolution setting, we obtain the best attainable global rates of convergence. With the existing techniques, it would be extremely difficult to solve such a problem. oAbbreviated title. Local I-d subproblems. AMS 1980 subject classification. Primary 62G20. Secondary 62G05.