Uniform-in-bandwidth consistency results in the partially linear additive model components estimation

Abstract

We are mainly concerned with the partially linear additive model defined for a measurable function $\psi :{\mathbb{R}}^q\to \mathbb{R},$ by

$$\psi ({\mathbf{Y}}_i):={\mathcal{Y}}_i={\mathbf{Z}}_i^{\top }\mathit{\beta }+\sum _{\mathcal{l}=1}^d{m}_{\mathcal{l}}(X_{\mathcal{l},i})+{\epsilon }_i \mathrm{ }\text{for}\mathrm{ }1\le i\le n,$$

where ${\mathbf{Z}}_i={(Z_{1,i},\dots ,Z_{p,i})}^{\top }$ and ${\mathbf{X}}_i={(X_{1,i},\dots ,X_{i,d})}^{\top }$ are vectors of explanatory variables, $\mathit{\beta }={({\beta }_1,\dots ,{\beta }_p)}^{\top }$ is a vector of unknown parameters, $m_1,\dots ,m_d$ are unknown univariate real functions, and ${\epsilon }_1,\dots ,{\epsilon }_n$ are independent random errors with mean zero, finite variances ${\sigma }_{\epsilon }$ and $\mathbb{E}(\epsilon |\mathbf{X},\mathbf{Z})=0$ a.s. Under some mild conditions, we present a sharp uniform-in-bandwidth limit law for the nonlinear additive components of the model estimated by the marginal integration device with the kernel method. We allow the bandwidth to varying within the complete range for which the estimator is consistent. We provide the almost sure simultaneous asymptotic confidence bands for the regression functions.

Keywords:

• Additive model
• regression function
• marginal integration
• non-parametric estimation
• density estimation
• regression estimation

AMS SUBJECT CLASSIFICATIONS:

• 62G20
• 62G32
• 62J05
• 60G07
• 60F15
• 62G08

Acknowledgments

The author would like to thank the Editor-in-Chief, an Associate-Editor, and two referees for their extremely helpful remarks, which resulted in a substantial improvement of the original form of the work and a presentation that was more sharply focused.

CRediT author statement

Khalid Chokri: Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing.

Salim Bouzebda: Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing.

Both authors contributed equally to this work.

Disclosure statement

No potential conflict of interest was reported by the authors.