Asymptotic results for the linear parameter estimate in partially linear additive regression model

Abstract

In this Note, we study the linear part of the semi-parametric regression model defined by Y i = Z i ⊤ β + ∑ j = 1 d m j ( X i j ) + ε i , 1 ⩽ i ⩽ n , where Z i = ( Z i 1 , … , Z i p ) ⊤ , X i = ( X i 1 , … , 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 modelling errors with mean zero and finite variances. Using the nonparametric kernel technique combined with the marginal integration method to estimate the functions ( m j ) 1 ⩽ j ⩽ d and the least-square error criterion to estimate the parameter β, we establish the asymptotic normality together with the iterated logarithm law of the estimate β ˆ of β.