A B-spline robust approach for partially linear additive models

MARTINEZ, ALEJANDRA; BOENTE, GRACIELA

Congreso; The International Conference on Robust Statistics; 2021

To deal with the curse of dimensionality, partially linear additive models provide a flexible and interpretable approach to build predictive models. They combine features in an additive manner, allowing each of them to have either a linear or non?linear effect on the response. More precisely, under a partially linear additive model both parametric andnonparametric components coexist. Among others, [2] describes some of the advantages of partially linear additive models, including the facts that they are easily interpretable, and that the estimators for the parametric components are more efficient. Under a partially linear additive model, we deal with independent and identically distributed observations (Yi, Zi, Xi) ∈ IR × IR^q × IR^p , 1 ≤ i ≤ n, such thatY i = µ + β Zi +\sum_{j=1}^p η j (X_ij ) + σεi ,where Xi = (X_i1 , . . . , X_ip) and the errors εi are centered and independent from the covariates. Then, the univariate unknown functions η_j : IR → IR (1 ≤ j ≤ p), the coefficients µ ∈ IR, β ∈ IR^q and the scale parameter σ > 0 are the quantities to be estimated. Classical estimation procedures based on least squares assume that IE(εi) = 0and Var(εi) = 1 and may be found in [1], for instance.In this presentation, we introduce a family of robust estimators that combines B-splines with robust MM-regression estimators and avoids moment conditions for the errors. Consistency results, rates of convergence of the robust estimators as well as the asymptotic distribution of the estimators of β are obtained under mild assumptions. To select the dimension of the B-spline basis, a robust version of the BIC criteria is considered. Through the results of a Monte Carlo experiment we will show the advantage of the proposed methodology over the classical one for finite samples. Finally, we will also illustrate the robust proposal on a real data set.