Mathematics – Statistics Theory
Scientific paper
2009-12-22
Mathematics
Statistics Theory
31 pages
Scientific paper
The paper concerns the problem of pointwise adaptive estimation in regression when the noise is heteroscedastic and incorrectly known. We use the method of local approximation including as a particular case the local polynomial smoothing. Specifically, the model with unknown mean and variance is approximated by a local linear model with an incorrectly specified covariance matrix. Adaptive choice of degree of localization in this case can be understood as a choice of an appropriate parametric model from a given collection. For the selection from the family of models we employ based on Lepski's method the FLL technique recently suggested in Katkovnik and Spokoiny (2008). The problem of the choice of certain parameters in this type of procedures was addressed in Spokoiny and Vial (2009). The authors called their approach to the calibration of the parameters "propagation". We developed and justified the methodology for the heteroscedastic case in the presence of noise misspecification. The analysis shows that the adaptive procedure allows a misspecification of the covariance matrix with a relative error of order 1/log n, where n is the sample size.
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