Adaptive estimation under misspecification in regression

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 (2009). The "propagation conditions" are relaxed for the model with misspecified covariance structure. Specifically, the model with unknown mean and variance is approximated by the one with parametric assumption of local linearity of the mean function and with an incorrectly specified covariance matrix. We show that this procedure allows 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 nonasymptotic "oracle" risk bounds.