Title: Nearly Semiparametric Efficient Estimation of Quantile Regression,
Abstract: As a competitive alternative to least squares regression, quantile regression
is popular in analyzing heterogenous data. For quantile regression model
specified for one single quantile level $\tau$, major difficulties of
semiparametric efficient estimation are the unavailability of a parametric
efficient score and the conditional density estimation. In this paper, with the
help of the least favorable submodel technique, we first derive the
semiparametric efficient scores for linear quantile regression models that are
assumed for a single quantile level, multiple quantile levels and all the
quantile levels in $(0,1)$ respectively. Our main discovery is a one-step
(nearly) semiparametric efficient estimation for the regression coefficients of
the quantile regression models assumed for multiple quantile levels, which has
several advantages: it could be regarded as an optimal way to pool information
across multiple/other quantiles for efficiency gain; it is computationally
feasible and easy to implement, as the initial estimator is easily available;
due to the nature of quantile regression models under investigation, the
conditional density estimation is straightforward by plugging in an initial
estimator. The resulting estimator is proved to achieve the corresponding
semiparametric efficiency lower bound under regularity conditions. Numerical
studies including simulations and an example of birth weight of children
confirms that the proposed estimator leads to higher efficiency compared with
the Koenker-Bassett quantile regression estimator for all quantiles of
interest.