Abstract
The paper concerns the problem of pointwise adaptive estimation in regression when the noise is heteroscedastic and incorrectly known. The use of the local approximation method, which includes the local polynomial smoothing as a particular case, leads to a finite family of estimators corresponding to different degrees of smoothing. Data-driven choice of localization degree in this case can be understood as the problem of selection from this family. This task can be performed by a suggested in Katkovnik and Spokoiny (2008) FLL technique based on Lepski's method. An important issue with this type of procedures - the choice of certain tuning parameters - was addressed in Spokoiny and Vial (2009). The authors called their approach to the parameter calibration "propagation". In the present paper the propagation approach is developed and justified for the heteroscedastic case in presence of the noise misspecification. Our 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.
Highlights
Consider a regression modelY = f + Σ10/2ε, ε ∼ N (0, In) (1.1)with response vector Y ∈ Rn and unknown diagonal covariance matrix Σ0 = diag(σ02,1, . . . , σ02,n)
The use of the local approximation method, which includes the local polynomial smoothing as a particular case, leads to a finite family of estimators corresponding to different degrees of smoothing
Data-driven choice of localization degree in this case can be understood as the problem of selection from this family. This task can be performed by a suggested in Katkovnik and Spokoiny (2008) fitted local likelihood (FLL) technique based on Lepski’s method
Summary
The local constant fit at a given point x ∈ IR is covered as well with p = 1 In this case the “design” matrix is a row Ψ = The problem of “local model selection” addressed in the present paper is quite different to the model selection in the sense of [5] and [33] related to estimation with global risk In this set-up an amazing progress is achieved for the model selection in heteroscedastic not necessary Gaussian regression model in [2], [3], [35]. The minimax pointwise estimation in heteroscedastic regression is in focus of [7]
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