Bootstrap approximation in a partially linear regression model

Consider the semiparametric regression model Y-i= X(i)(T)beta +g(T-i)+epsilon (i) (i = 1,...,n), where (X-i, T-i) are known and fixed design points, beta is a p-dimensional unknown parameter, g(.) is an unknown function on [0, 1], and epsilon (i) are i.i.d. random errors with mean 0 and variance sigma (2). In this paper, we first construct bootstrap statistics beta (*)(n) and sigma (2)(n) by resampling. Then we prove that far the estimators beta (n) and sigma (2)(n) of the parameters beta and sigma (2), rootn(beta (*)(n) - beta (n)) and rootn(beta (n) - beta), rootn(sigma (2*)(n) - sigma (2)(n)) and rootn(sigma (2)(n) - sigma (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. (C) 2000 Elsevier Science B.V. All rights reserved.