Likelihood Based Inference for Quantile Regression Using the Asymmetric Laplace Distribution

To make inferences about the shape of a population distribution, the widely p opular mean regression model, for example, is inadequate if the distribution is not approx imately Gaussian (or symmetric). Compared to conventional mean regression (MR), quantile r egr ssion (QR) can characterize the entire conditional distribution of the outcome variable, a nd is more robust to outliers and misspecification of the error distribution. We present a lik elihood-based approach to the estimation of the regression quantiles based on the asymmetric L apla e distribution (ALD), a choice that turns out to be natural in this context. The ALD h as a nice hierarchical representation which facilitates the implementation of the EM algorithm for ma ximumlikelihood estimation of the parameters at the pth level with the observed information matrix as a byproduct. Inspired by the EM algorithm, we develop case-deletion dia g ostics analysis for QR models, following the approach of Zhu et al. (2001). This is becau se the observed data log–likelihood function associated with the proposed model is somewhat c omplex (e.g., not differentiable at zero) and by using Cook’s well-known approach it an be very difficult to obtain case-deletion measures. The techniques are illustrated with both simula ted and real data. In particular, in an empirical comparison, our approach out-perfo rmed other common classic estimators under a wide array of simulated data models and is flexible eno ugh to easily accommodate changes in their assumed distribution. The proposed algorithm a nd methods are implemented in the R package ALDqr().