Noncrossing structured additive multiple-output Bayesian quantile regression models

Quantile regression models are a powerful tool for studying different points of the conditional distribution of univariate response variables. Their multivariate counterpart extension though is not straightforward, starting with the definition of multivariate quantiles. We propose here a flexible Bayesian quantile regression model when the response variable is multivariate, where we are able to define a structured additive framework for all predictor variables. We build on previous ideas considering a directional approach to define the quantiles of a response variable with multiple outputs, and we define noncrossing quantiles in every directional quantile model. We define a Markov chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing property is obtained considering a Gaussian process design to model the correlation between several quantile regression models. We illustrate the results of these models using two datasets: one on dimensions of inequality in the population, such as income and health; the second on scores of students in the Brazilian High School National Exam, considering three dimensions for the response variable.

[1]  V. Chernozhukov,et al.  Massachusetts Institute of Technology Department of Economics Working Paper Series Improving Point and Interval Estimates of Monotone Functions by Rearrangement Improving Point and Interval Estimates of Monotone Functions by Rearrangement , 2022 .

[2]  J. S. Silva,et al.  Quantiles for Counts , 2002 .

[3]  D. Paindaveine,et al.  Local constant and local bilinear multiple-output quantile regression , 2012 .

[4]  D. Paindaveine,et al.  Local bilinear multiple-output quantile/depth regression , 2015, 1507.07754.

[5]  M. Guggisberg.  A Bayesian Approach to Multiple-Output Quantile Regression , 2019, Journal of the American Statistical Association.

[6]  Keming Yu,et al.  Bayesian quantile regression , 2001 .

[7]  P. Rousseeuw,et al.  Halfspace Depth and Regression Depth Characterize the Empirical Distribution , 1999 .

[8]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[9]  Yanan Fan,et al.  Simultaneous fitting of Bayesian penalised quantile splines , 2019, Comput. Stat. Data Anal..

[10]  Luping Zhao,et al.  A Bayesian Semiparametric Temporally-Stratified Proportional Hazards Model with Spatial Frailties. , 2012, Bayesian analysis.

[11]  Robert Serfling,et al.  Quantile functions for multivariate analysis: approaches and applications , 2002 .

[12]  C. Schmitt,et al.  Socio-Economic Panel (SOEP), data from 1984-2014 (international Version) , 2016 .

[13]  Keming Yu,et al.  A Three-Parameter Asymmetric Laplace Distribution and Its Extension , 2005 .

[14]  Yanan Fan,et al.  Regression Adjustment for Noncrossing Bayesian Quantile Regression , 2015, 1502.01115.

[15]  T. McCowan.  Expansion without equity: An analysis of current policy on access to higher education in Brazil , 2007 .

[16]  J. Kadane,et al.  Simultaneous Linear Quantile Regression: A Semiparametric Bayesian Approach , 2012 .

[17]  Yu Ryan Yue,et al.  For a list of recent papers see the backpages of this paper. Bayesian Semiparametric Additive Quantile Regression , 2022 .

[18]  R. Ramamoorthi,et al.  Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density , 2013 .

[19]  Mitchell J. Small,et al.  Impact of Beliefs About Atlantic Tropical Cyclone Detection on Conclusions About Trends in Tropical Cyclone Numbers , 2011 .

[20]  D. Paindaveine,et al.  Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth , 2010, 1002.4486.

[21]  M. Hallin,et al.  Multiple-Output Quantile Regression , 2016 .

[22]  Nadja Klein,et al.  Simultaneous inference in structured additive conditional copula regression models: a unifying Bayesian approach , 2016, Stat. Comput..

[23]  R. Koenker.  Quantile Regression: Name Index , 2005 .

[24]  S. Klasen,et al.  Reconsidering the income‐health relationship using distributional regression , 2018, Health economics.

[25]  R. Koenker,et al.  Goodness of Fit and Related Inference Processes for Quantile Regression , 1999 .

[26]  H. Kozumi,et al.  Gibbs sampling methods for Bayesian quantile regression , 2011 .

[27]  Guillaume Carlier,et al.  Vector quantile regression beyond the specified case , 2017, J. Multivar. Anal..

[28]  Yun Yang,et al.  Joint Estimation of Quantile Planes Over Arbitrary Predictor Spaces , 2015, 1507.03130.

[29]  Stefan Lang,et al.  For a list of recent papers see the backpages of this paper. Multilevel , 2022 .

[30]  R. Koenker.  Quantile Regression: Fundamentals of Quantile Regression , 2005 .

[31]  G. Carlier,et al.  Vector Quantile Regression , 2014, 1406.4643.

[32]  H. Bondell,et al.  Noncrossing quantile regression curve estimation. , 2010, Biometrika.

[33]  Xuming He.  Quantile Curves without Crossing , 1997 .

[34]  Linglong Kong,et al.  Quantile tomography: using quantiles with multivariate data , 2008, Statistica Sinica.

[35]  Keming Yu,et al.  Quantile regression: applications and current research areas , 2003 .

[36]  G. Carlier,et al.  Vector Quantile Regression , 2014 .

[37]  Nan Lin,et al.  Model selection in binary and tobit quantile regression using the Gibbs sampler , 2012, Comput. Stat. Data Anal..

[38]  Miroslav Siman,et al.  Computing multiple-output regression quantile regions , 2012, Comput. Stat. Data Anal..

[39]  M. Fuentes,et al.  Journal of the American Statistical Association Bayesian Spatial Quantile Regression Bayesian Spatial Quantile Regression , 2022 .

[40]  Pavlo Mozharovskyi,et al.  Exact computation of the halfspace depth , 2014, Comput. Stat. Data Anal..

[41]  V. Chernozhukov,et al.  Monge-Kantorovich Depth, Quantiles, Ranks and Signs , 2014, 1412.8434.

[42]  J.-L. Dortet-Bernadet,et al.  Pyramid Quantile Regression , 2016, Journal of Computational and Graphical Statistics.