Title: Bootstrap of residual processes in regression: to smooth or not to smooth ?,
Abstract: In this paper we consider a location model of the form $Y = m(X) +
\varepsilon$, where $m(\cdot)$ is the unknown regression function, the error
$\varepsilon$ is independent of the $p$-dimensional covariate $X$ and
$E(\varepsilon)=0$. Given i.i.d. data $(X_1,Y_1),\ldots,(X_n,Y_n)$ and given an
estimator $\hat m(\cdot)$ of the function $m(\cdot)$ (which can be parametric
or nonparametric of nature), we estimate the distribution of the error term
$\varepsilon$ by the empirical distribution of the residuals $Y_i-\hat m(X_i)$,
$i=1,\ldots,n$. To approximate the distribution of this estimator, Koul and
Lahiri (1994) and Neumeyer (2008, 2009) proposed bootstrap procedures, based on
smoothing the residuals either before or after drawing bootstrap samples. So
far it has been an open question whether a classical non-smooth residual
bootstrap is asymptotically valid in this context. In this paper we solve this
open problem, and show that the non-smooth residual bootstrap is consistent. We
illustrate this theoretical result by means of simulations, that show the
accuracy of this bootstrap procedure for various models, testing procedures and
sample sizes.