Likelihood Based Inference for Quantile Regression Using the Asymmetric Laplace Distribution

Luis Benites Sánchez
Víctor H. Lachos
Filidor E. Vilca-Labra

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), a choice that turns out to be natural in this context. The ALD has a nice hierarchical representation which facilitates the implementation of the EM algorithm for maximum-likelihood 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 diagnostics analysis for QR models, following the approach of Zhu et al. (2001). This is because the observed data log–likelihood function associated with the proposed model is somewhat complex (e.g., not differentiable at zero) and by using Cook’s well-known approach it can be very difficult to obtain case-deletion measures. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed other common classic estimators under a wide array of simulated data models and is flexible enough to easily accommodate changes in their assumed distribution. The proposed algorithm and methods are implemented in the R package ALDqr().

Quantile regression model
EM algorithm
Case-deletion model
asymmetric Laplace distribution