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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 anestimator $\hat m(\cdot)$ of the function $m(\cdot)$ (which can be parametricor 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 andLahiri (1994) and Neumeyer (2008, 2009) proposed bootstrap procedures, based onsmoothing the residuals either before or after drawing bootstrap samples. Sofar it has been an open question whether a classical non-smooth residualbootstrap is asymptotically valid in this context. In this paper we solve thisopen problem, and show that the non-smooth residual bootstrap is consistent. Weillustrate this theoretical result by means of simulations, that show theaccuracy of this bootstrap procedure for various models, testing procedures andsample sizes.

Bootstrap of residual processes in regression: to smooth or not to smooth ?