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().