  Quantile regression is a technique to estimate conditional quantile curves.
It provides a comprehensive picture of a response contingent on explanatory
variables. In a flexible modeling framework, a specific form of the conditional
quantile curve is not a priori fixed. % Indeed, the majority of applications do
not per se require specific functional forms. This motivates a local parametric
rather than a global fixed model fitting approach. A nonparametric smoothing
estimator of the conditional quantile curve requires to balance between local
curvature and stochastic variability. In this paper, we suggest a local model
selection technique that provides an adaptive estimator of the conditional
quantile regression curve at each design point. Theoretical results claim that
the proposed adaptive procedure performs as good as an oracle which would
minimize the local estimation risk for the problem at hand. We illustrate the
performance of the procedure by an extensive simulation study and consider a
couple of applications: to tail dependence analysis for the Hong Kong stock
market and to analysis of the distributions of the risk factors of temperature
dynamics.
