Local parametric estimation under noise misspecification in regression

The problem of pointwise estimation for nonparametric regression with heteroscedastic additive Gaussian noise is considered. The approach is an adaptive estimation with application of Lepski's method to selecting of one estimate from the set of linear estimates obtained by different degrees of localization. This approach is combined with the "propagation conditions" on the choice of the critical values of the procedure under the simplest null hypothesis suggested recently by V. Spokoiny in his joint work with C. Vial. We study a general linear parametric approximation to the mean function together with relaxation of conditions on the noise. We also consider the "propagation conditions" for the model with unknown covariance structure. This means that the covariance matrix is supposed to be wrongly known implying "noise misspecification". The model with unknown mean and variance is approximated by the one with parametric assumption of local linearity of the mean function and with a wrong covariance matrix. An analysis of this procedure allows for a misspecification of the covariance matrix with a relative error up to o(1/log(n)), where n is the sample size. The quality of estimation is measured in terms of non-asymptotic "oracle" risk bounds.