Asymptotic normality for the wavelet partially linear additive model components estimation

Abstract

The focus of this article is on studying a partially linear additive model, which is defined using a measurable function ψ : R q → R . The model is given as follows: ψ ( Y i ) := Y i = Z i ⊤ β + ∑ ℓ = 1 d m ℓ ( X ℓ , i ) + ε i for 1 ≤ i ≤ n , where Z i = ( Z i , 1 , … , Z i p ) ⊤ and X i = ( X 1 , i , … , X i d ) ⊤ are vectors of explanatory variables, β = ( β 1 , … , β p ) ⊤ is a vector of unknown parameters, m 1 , … , m d are unknown univariate real functions, and ε 1 , … , ε n are independent random errors with mean zero, finite variances σ ε . Additionally, it is assumed that E ( ε | X , Z ) = 0 almost surely. The main contributions of this article are as follows. First, under certain mild conditions, we establish the asymptotic normality of the non linear additive components of the model. These components are estimated using the marginal integration device with the linear wavelet method. Second, we leverage our main result to construct confidence intervals for the estimated model.