Statistics and Its Interface

Volume 13 (2020)

Number 2

Sparse signal shrinkage and outlier detection in high-dimensional quantile regression with variational Bayes

Pages: 237 – 249



Daeyoung Lim (Department of Statistics, University of Connecticut, Storrs, Ct., U.S.A.)

Beomjo Park (Department of Statistics, Carnegie Mellon University, Pittsburg, Pennsylvania, U.S.A.)

David Nott (Department of Statistics and Applied Probability, National University of Singapore)

Xueou Wang (Department of Statistics and Applied Probability, National University of Singapore)

Taeryon Choi (Department of Statistics, Korea University, Seoul, South Korea)


Model misspecification can compromise valid inference in conventional quantile regression models. To address this issue, we consider two flexible model extensions for high-dimensional data. The first is a Bayesian quantile regression approach with variable selection, which uses a sparse signal shrinkage prior on the high-dimensional regression coefficients. The second extension robustifies conventional parametric quantile regression methods by including observation specific mean shift terms. Since the number of outliers is assumed to be small, the vector of mean shifts is sparse, which again motivates the use of a sparse signal shrinkage prior. Specifically, we exploit the horseshoe+ prior distribution for variable selection and outlier detection in the high-dimensional quantile regression models. Computational complexity is alleviated using fast mean field variational Bayes methods, and we compare results obtained by variational methods with those obtained using Markov chain Monte Carlo (MCMC).


Asymmetric Laplace distribution, Horseshoe+ prior, Outlier detection, Quantile regression, Variational Bayes

2010 Mathematics Subject Classification

62J07, 62F15, 62G35

The research of Taeryon Choi was supported by a Korea University Grant (K1807851), and by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2019R1A2C1010018).

Received 3 March 2019

Accepted 20 November 2019

Published 30 January 2020