  In this thesis we study adaptive nonparametric regression with noise
misspecification and the complexity of approximation of random fields in
dependence of the dimension.
  First, we consider the problem of pointwise estimation in nonparametric
regression with heteroscedastic additive Gaussian noise. We use the method of
local approximation applying the Lepski method for selecting one estimate from
the set of linear estimates obtained by the different degrees of localization.
This approach is combined with the "propagation conditions" on the choice of
critical values of the procedure, as suggested recently by Spokoiny and Vial
[Ann.Stat., 2009]. The "propagation conditions" are relaxed for the model with
misspecified covariance structure. We show that this procedure allows a
misspecification of the covariance matrix with a relative error of order 1/
log(n), where n is the sample size. The quality of estimation is measured in
terms of "oracle" risk bounds.
  We then turn to the approximation of d-parametric random fields of tensor
product-type by means of n-term partial sums of the Karhunen-Lo\`eve expansion.
The analysis is restricted to the average case setting. The quantity of
interest is the information complexity describing the minimal number of terms
in the partial sums, which guarantees an error not exceeding a given level. The
behavior of this quantity when the dimension goes to infinity is the subject of
our study.
