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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 σε. 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. [ABSTRACT FROM AUTHOR]

We are mainly concerned with the partially linear additive model defined for a measurable function ψ : R q → R , by 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 1 , i , ... , Z p , i) ⊤ 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 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. [ABSTRACT FROM AUTHOR]

In the present paper, we are mainly concerned with the partially linear additive model defined, for a measurable function, bywhereZi= (Zi, 1, …,Zip)⊤andXi= (X1, i, …,Xid)⊤are vectors of explanatory variables,is a vector of unknown parameters,m1, …,mdare unknown univariate real functions, and ϵ1, …, ϵnare independent random errors with mean zero, finite variances σϵ, anda.s. We establish exact rates of strong uniform consistency of the non linear additive components of the model estimated by the marginal integration device with the kernel method. Our proofs are based upon the modern empirical process theory in the spirit of the works of Einmahl and Mason (2000) and Deheuvels and Mason (2004) relative to uniform deviations of non parametric kernel-type estimators. [ABSTRACT FROM PUBLISHER]