Bootstrap approximation in a partially linear regression model

Abstract

Abstract Consider the semiparametric regression model Y i =X i T β+g(T i )+e i (i=1,…,n), where ( X i , T i ) are known and fixed design points, β is a p -dimensional unknown parameter, g (·) is an unknown function on [0,1], and e i are i.i.d. random errors with mean 0 and variance σ 2 . In this paper, we first construct bootstrap statistics β n ∗ and σ n 2 ∗ by resampling. Then we prove that for the estimators β n and σ n 2 of the parameters β and σ 2 , n (β n ∗ −β n ) and n (β n −β), n (σ n 2 ∗ −σ n 2 ) and n (σ n 2 −σ 2 ) have the same limit distributions, respectively. The advantage of the bootstrap approximation is explained. The feasibility of this approach is also shown in a simulation study.