  In practical applications, one often does not know the "true" structure of
the underlying conditional quantile function, especially in the ultra-high
dimensional setting. To deal with ultra-high dimensionality, quantile-adaptive
marginal nonparametric screening methods have been recently developed. However,
these approaches may miss important covariates that are marginally independent
of the response, or may select unimportant covariates due to their high
correlations with important covariates. To mitigate such shortcomings, we
develop a conditional nonparametric quantile screening procedure (complemented
by subsequent selection) for nonparametric additive quantile regression models.
Under some mild conditions, we show that the proposed screening method can
identify all relevant covariates in a small number of steps with probability
approaching one. The subsequent narrowed best subset (via a modified Bayesian
information criterion) also contains all the relevant covariates with
overwhelming probability. The advantages of our proposed procedure are
demonstrated through simulation studies and a real data example.
