Optimal Almost Sure Rate of Convergence for the Wavelets Estimator in the Partially Linear Additive Models

In this article, we examine a class of partially linear additive models (PLAM) defined via a measurable mapping Ψ:Rq→R. More precisely, we consider Ψ(Yi):=Yi=Zi⊤β+∑l=1dml(Xl,i)+εi,i=1,…,n, where Zi=(Zi,1,…,Zi,p)⊤ and Xi=(X1,i,…,Xd,i)⊤ denote vectors of explanatory variables. The unknown parameter vector is β=(β1,…,βp)⊤, and m1,…,md are real-valued functions of a single variable whose forms are not specified. The error terms ε1,…,εn are identically distributed with mean zero and finite variance σε, and they fulfill the condition E(ε∣X,Z)=0 almost surely. These models are broadly applicable in finance, biology, and engineering, where capturing intricate nonlinear effects is essential. We propose an estimation method that leverages marginal integration in conjunction with linear wavelet-based techniques to obtain estimators for the unknown components m1,…,md. Under suitable regularity conditions, we establish strong uniform convergence of these estimators, demonstrating that they achieve practically relevant convergence rates. Our theoretical results indicate that these estimators converge uniformly at rates that are favorable for practical applications, underscoring the adaptability and scope of this partially linear additive model.