The known asymptotically efficient estimates of a nonlinear functional of an unknown function are crucially depended on prior information about a smoothness of this function. We show on example of filtering that this information is not necessary at all whenever the asymptotically efficient...

In a controlled random design experiment it is often desirable to estimate an underlying regression function with assigned risk, then the optimality is understood as a minimal mean sample size. For the first time in the literature, it is proved that such a sequential procedure exists. Further,...

In a non-parametric regression, the heteroscedasticity (dependence of the variance of the regression error on the predictor) can be a serious complication in estimation or visualization of an underlying regression function. If a controlled sampling is permitted, then the statistician can choose...

One of the common themes of the modern nonparametric functional estimation folklore is that the signal-in-white-noise model can serve as a prototype for nonparametric regression. In this note we give an example showing that it may not always be the case

The results of Hall et al. (1998, Ann. Statist. 26, 922-943) together with Efromovich (2000, Bernoulli) imply that a data-driven block shrinkage wavelet estimator, which mimics a sharp minimax linear oracle, is rate optimal over spatially inhomogeneous function spaces. This result does not...

Results on sharp minimax estimation of the probability density from a Sobolev function class, established in the 1980s, were among the first sharp minimax results in the nonparametric curve estimation literature. This paper considers the last unsolved case -- a Sobolev class of order of...