Adaptive quantile regression

Sara van de Geer

In quantile regression, one tries to find the
relation g(x) between covariables x on the one hand 
and a certain quantile of the distribution
of a response variable y on the other hand. 
An important special case is the median of y, which corresponds
to the least absolute deviations loss function.

We consider the case where the regression function g(x)
is estimated using a complexity penalty on the quantile loss function.
Under standard identifiability conditions, one may derive
oracle inequalities for the penalized quantile
regression estimator. 

Our focus point will be the identifiability condition.
By standard identifiability, we mean that the theoretical
loss behaves quadratically near its minimizer.
However, in some situations the behavior is
a power k larger than 2, which means that the
true regression is harder to identify.
This occurs for instance when densities of the measurement 
error vanish at the quantile. Another situation where
this can occur is the case where the class of regression functions
allowed is not a linear space.

Possible violations of the standard identifiability
assumption (k=2) may have a great impact on the behavior of an
estimator. In fact, penalties that work well
(give adaptive estimators) in the standard case may become
inconsistent in the non standard case (k>2).

We will show that a soft thresholding type penalty
allows one to adapt to possible violations of the standard
identifiability conditions. This estimator
neither  requires knowledge of k, nor knowledge of the smoothness
of the regression. Furthermore, the estimator can be computed
using a standard quantile regression fitting routine.