1. Bootstrap of residual processes in regression: to smooth or not to smooth?
- Author
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I Van Keilegom, Natalie Neumeyer, and UCL - SSH/LIDAM/ISBA - Institut de Statistique, Biostatistique et Sciences Actuarielles
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Independent and identically distributed random variables ,Statistics::Theory ,General Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Residual ,01 natural sciences ,Methodology (stat.ME) ,010104 statistics & probability ,0502 economics and business ,FOS: Mathematics ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Statistics - Methodology ,050205 econometrics ,Mathematics ,Applied Mathematics ,05 social sciences ,Nonparametric statistics ,Estimator ,Regression analysis ,Agricultural and Biological Sciences (miscellaneous) ,Empirical distribution function ,Nonparametric regression ,Kernel smoother ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences - 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.
- Published
- 2019
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