  Quantile regression is a tool for learning conditional distributions. In this
paper we study quantile regression in the setting where a protected attribute
is unavailable when fitting the model. This can lead to "unfair'' quantile
estimators for which the effective quantiles are very different for the
subpopulations defined by the protected attribute. We propose a procedure for
adjusting the estimator on a heldout sample where the protected attribute is
available. The main result of the paper is an empirical process analysis
showing that the adjustment leads to a fair estimator for which the target
quantiles are brought into balance, in a statistical sense that we call
$\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a
dataset of 200,000 live births, where the objective is to characterize the
dependence of the birth weights of the babies on demographic attributes of the
birth mother; the protected attribute is the mother's race.
