A Robust and Sparse Approach in Partially Linear Additive Models

• Departmento de Ciencias Básicas, Universidad Nacional de Luján and CONICET, Argentina [ammartinez@conicet.gov.ar]

Keywords: Sparsity – Robustness – Regularization

1 Abstract

Partially linear additive models (plam) assume that ${(Y_i,{\text{𝐙}}_i^{\mathrm{\text{t}}},\text{𝐗}{i}^{\mathrm{\text{t}}})}^{\mathrm{\text{t}}}\in {ℝ}^{1+q+p}$, $1\le i\le n$, are i.i.d. random vectors following

$$Y=m({\text{𝐙}}^{\mathrm{\text{t}}},{\text{𝐗}}^{\mathrm{\text{t}}})+u=\mu +{𝜷}^{\mathrm{\text{t}}}\text{𝐙}+\sum _{j=1}^p{\eta }_j(X_j)+\sigma \epsilon .$$

where $\mu \in ℝ$, $𝜷\in {ℝ}^q$, ${\eta }_j:ℝ\to ℝ$ satisfy $\int {\eta }_j(x)𝑑x=0$, and $\sigma >0$. The errors $\epsilon$ are independent of ${({\text{𝐙}}^{\mathrm{\text{t}}},{\text{𝐗}}^{\mathrm{\text{t}}})}^{\mathrm{\text{t}}}$.

Usually, in a first step of modelling, researchers introduce all the available variables in the model and so those that have a small impact on the response variable will reduce the prediction capability of the estimators. For this reason, variable selection plays an important role. This talk will consider a regularization procedure for variable selection.

We will present a robust method for simultaneous estimation and selection in sparse plams, enhancing resistance to outliers while preserving model sparsity. Additionally, we will discuss a method for automatically selecting the penalty parameters.

Through a simulation study, we will compare the robust proposal with its least-squares counterpart in a high-dimensional setting. The results hightlight the stability of the robust proposal and its advantage in handling atypical data while performing automatic variable selection.