  To make inferences about the shape of a population distribution, the widely
popular mean regression model, for example, is inadequate if the distribution
is not approximately Gaussian (or symmetric). Compared to conventional mean
regression (MR), quantile regression (QR) can characterize the entire
conditional distribution of the outcome variable, and is more robust to
outliers and misspecification of the error distribution. We present a
likelihood-based approach to the estimation of the regression quantiles based
on the asymmetric Laplace distribution (ALD), which has a hierarchical
representation that facilitates the implementation of the EM algorithm for the
maximum-likelihood estimation. We develop a case-deletion diagnostic analysis
for QR models based on the conditional expectation of the complete-data
log-likelihood function related to the EM algorithm. The techniques are
illustrated with both simulated and real data sets, showing that our approach
out-performed other common classic estimators. The proposed algorithm and
methods are implemented in the R package ALDqr().
