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 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.