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

Khalid Chokri and Salim Bouzebda

Communications in Statistics - Theory and Methods, 2024, vol. 53, issue 9, 3383-3424

Abstract: We are mainly concerned with the partially linear additive model defined for a measurable function ψ:Rq→R, byψ(Yi):=Yi=Zi⊤β+∑l=1dml(Xl,i)+εi for 1≤i≤n,where Zi=(Z1,i,…,Zp,i)⊤ and Xi=(X1,i,…,Xi,d)⊤ are vectors of explanatory variables, β=(β1,…,βp)⊤ is a vector of unknown parameters, m1,…,md are unknown univariate real functions, and ε1,…,εn are independent random errors with mean zero, finite variances σε and E(ε|X,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.

Date: 2024
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DOI: 10.1080/03610926.2022.2153605

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