It is known that both Watson-Nadaraya and Gasser-Muller types of regression estimators have some disadvantages. A smooth version of local polynomial regression estimators are proposed to remedy the disadvantages. The mean squared error and mean integrated squared errors are computed explicitly. It turns out that by suitably selecting a kernel and a bandwidth, the proposed estimator has at least asymptotic minimax efficiency 89.6%-proposed estimator is efficient in rates and nearly efficient in constant factors! In nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of the "hardest 1-dimensional subproblem". The explicit connections of minimax risks with modulus of continuity are made. Normal submodels are used to avoid the technical difficulty of Le Cam's theory of convergence of experiments. The lower bound is applicable for estimating conditional mean (regression) and conditional quantiles (including median) for both fixed design and random design regression problems. Abbreviated title. Minimax nonparametric regression. AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62F35. Key words and phrases. Hardest I-dimensional subproblem; Local polynomials; Minimax risk; Modulus

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.