Estimation of additive quantile regression

Abstract

Conditional quantiles, Additive models, Marginal integration, Non-increasing rearrangements,

Testing for additivity in nonparametric quantile regression

“…Similarly, Q −i,j add and Q −i,j,k add denote the corresponding estimators without the ith and jth and the ith, jth and kth observation, respectively. Various additive quantile regression estimates have been proposed by De Gooijer and Zerom (2003), Horowitz and Lee (2005), Lee et al (2010) and Dette and Scheder (2011). Note that statistics of the type (2.4) have been introduced by Zheng (1996) in the context of testing for a specific parametric form in nonparametric regression, and since its introduction has found considerable interest in the context of goodness-of-fit tests [see Dette and von Lieres und Wilkau (2001) or Zhang and Dette (2004) among others].…”

“…Similarly, the estimateQ l,N (τ |x) is robust with respect to the choice of the bandwidth b n if it is chosen sufficiently small [see Dette et al (2006)]. The estimate (2.6) suffers from the curse of dimensionality if the dimension d of the predictor is large and for this reason Dette and Scheder (2011) propose to combine it with the marginal integration technique in order to obtain an additive estimate of the quantile regression with a one-dimensional rate of convergence. To be precise define…”

“…Moreover this model is a common assumption in quantile regression [see Hall et al (1999) or Dette and Scheder (2011) among others] and observing (3.4) we have…”

“…For all kernels in our estimators we use the Epanechnikov kernel K(t) = 3 4 (1 − t 2 )I [−1,1] (t), and a product of two kernels of this type as a two-dimensional kernel. Following Dette and Scheder (2011) the bandwidths are chosen as…”

“…Horowitz and Lee (2005) propose a two step procedure, which fits a parametric model in the first step (with increasing dimension) for each coordinate and smooth it in a second step by the local polynomial technique. Lee et al (2010) suggest backfitting methods for additive quantile regression estimation, while Dette and Scheder (2011) combine marginal integration techniques with monotone rearrangements [see Dette et al (2006)] for the construction of additive estimates. Although these methods estimate the unknown quantile regression with the optimal (one-dimensional) rate if the assumption of an additive model is correct, they are generally inconsistent if the quantile regression is not additive.…”