In this paper we consider a location model of the form $Y = m(X) +arepsilon$, where $m(cdot)$ is the unknown regression function, the error$arepsilon$ is independent of the $p$-dimensional covariate $X$ and$E(arepsilon)=0$. 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$arepsilon$ 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.
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机译：在本文中，我们考虑形式$ Y = M（X）+ varepsilon $，的位置的模型，其中$ M（ CDOT）$是未知回归函数，误差$ varepsilon $独立于$ P $的维协变量$ X $和$ E（ varepsilon）= 0 $。鉴于独立同分布数据$（X_1，Y_1）， ldots，（X_n，Y_n）$，并给出anestimator $ 帽子米（ CDOT）$的函数$ M（ CDOT）$（可parametricor非参数性质的），我们由残差$ Y_i- 帽子米（X_I）$的经验分布估计误差项$ varepsilon $的分布，$ I = 1， ldots，正$。为了接近这个估计之前onsmoothing的残差分布，Koul andLahiri（1994年）和Neumeyer（2008年，2009年）提出的自举程序，基于或绘图bootstrap样本后。 SOFAR它一直是一个悬而未决的问题是否经典的非平滑residualbootstrap就是在这种背景下渐近有效。在本文中，我们解决thisopen问题，表明非光滑剩余的引导是一致的。通过模拟的方式Weillustrate这一理论结果，显示各种型号自举程序theaccuracy，测试程序andsample大小。
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