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We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n y1 r2 . Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n y1 r2 . There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to magnetic resonance and satellite imaging.

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, This includes the "propagation" property which particularly yields the root-n consistency of the resulting estimate in the homogeneous case.We also state an "oracle" result which implies rate optimality of the estimate under usual smoothness conditions and a "separation" result which explains the sensitivity of the method to structural changes.