Joint estimation of quantile planes over arbitrary predictor spaces
This work addresses a key problem in statistics and econometrics for researchers and practitioners needing reliable quantile regression models, though it appears incremental as it builds on existing quantile regression frameworks with a new parametrization and Bayesian approach.
The paper tackles the challenge of jointly estimating non-crossing quantile planes over arbitrary convex predictor domains by proposing a novel parametrization that enables fast likelihood computation and Bayesian fitting. It introduces a Bayesian methodology with Gaussian process priors and MCMC estimation, showing posterior consistency and better accuracy, coverage, and model fit compared to existing methods in examples.
In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parametrization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit.