Workshop on Structure Adapting Methods - Abstract

Serdyukova, Nora

Local parametric estimation under noise misspecification in regression problem

The problem of pointwise estimation for local polynomial regression with heteroscedastic additive Gaussian noise is considered. The approach is an adaptive estimation with application of the Lepski's procedure selecting one estimate from the set of estimates obtained by different degrees of localization in combination with “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. The development of the Lepski-Spokoiny approach is firstly in consideration of rather general collection of localizing schemes, including as a particular case the popular kernel smoothing. Secondly, general polynomial approximation to the mean function is considered. The third and, probably, the main step forward is in relaxing the propagation approach to 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 ( obig( frac1log n big) ), where ( n ) is the sample size.