Distributed and parallel computing is becoming more important with the availability of extremely large data sets. In this article, we consider this problem for high-dimensional linear quantile regression. We work under the assumption that the coefficients in the regression model are sparse; therefore, a LASSO penalty is naturally used for estimation. We first extend the debiasing procedure, which is previously proposed for smooth parametric regression models to quantile regression. The technical challenges include dealing with the nondifferentiability of the loss function and the estimation of the unknown conditional density. In this article, the main objective is to derive a divide-and-conquer estimation approach using the debiased estimator which is useful under the big data setting. The effectiveness of distributed estimation is demonstrated using some numerical examples.

In this paper, the methods of information geometry are employed to investigate a generalized Bayes rule for prediction. Taking α-divergences as the loss functions, optimality and asymptotic properties of the generalized Bayesian predictive densities are considered. We show that the Bayesian predictive densities minimize a generalized Bayes risk. We also find that the asymptotic expansions of the densities are related to the coefficients of the α-connections of a statistical manifold. In addition, we discuss the difference between two risk functions of the generalized Bayesian predictions based on different priors. Finally, using the non-informative priors (i.e., Jeffreys and reference priors), uniform prior and conjugate prior, two examples are presented to illustrate the main results.