Joint Estimation of Quantile Planes Over Arbitrary Predictor Spaces

ABSTRACT

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 parameterization that characterizes any collection of noncrossing 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 parameterization 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. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian inference
• Bayesian nonparametric models
• Gaussian processes
• Joint quantile model
• Linear quantile regression

Supplementary Materials

The online supplement contains the appendices for the article.

Acknowledgements

The authors thank the Associate Editor and the referees for their suggestions to improving the article.

Funding

This research was supported by grants ES017436 and R01ES020619 from the National Institute of Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH).

Notes

¹As can be seen in the proof of Theorem 2, the function v(τ) of (5(5) $$\begin{array}{ccc}\hfill {\dot{\beta }}_0\left(\tau \right)>0,\dot{\beta }\left(\tau \right)& =& {\dot{\beta }}_0\left(\tau \right)\frac{v\left(\tau \right)}{a(v\left(\tau \right),\mathcal{X})\sqrt{1+{\parallel v\left(\tau \right)\parallel }^2}},\hfill \\ & & \tau \in (0,1),\hfill \end{array}$$(5) ) is expressed as w(ζ(τ)) in the reparameterization (6(6) $$\begin{array}{c}\hfill {\beta }_0\left({\tau }_0\right)={\gamma }_0,\beta \left({\tau }_0\right)=\gamma \end{array}$$(6) )–(8(8) $$\begin{array}{c}\hfill \beta \left(\tau \right)-\beta \left({\tau }_0\right)=\sigma {\int }_{\zeta \left({\tau }_0\right)}^{\zeta \left(\tau \right)}\frac{w\left(u\right)}{a(w\left(u\right),\mathcal{X})\sqrt{1+{\parallel w\left(u\right)\parallel }^2}}\\ \hfill \times q_0\left(u\right)du,\tau \in (0,1)\end{array}$$(8) ). This is not a restriction of any kind because ζ is a diffeomorphism. The particular reparameterization given in (6(6) $$\begin{array}{c}\hfill {\beta }_0\left({\tau }_0\right)={\gamma }_0,\beta \left({\tau }_0\right)=\gamma \end{array}$$(6) )–(8(8) $$\begin{array}{c}\hfill \beta \left(\tau \right)-\beta \left({\tau }_0\right)=\sigma {\int }_{\zeta \left({\tau }_0\right)}^{\zeta \left(\tau \right)}\frac{w\left(u\right)}{a(w\left(u\right),\mathcal{X})\sqrt{1+{\parallel w\left(u\right)\parallel }^2}}\\ \hfill \times q_0\left(u\right)du,\tau \in (0,1)\end{array}$$(8) ) is convenient for both theoretical calculations and numerical implementation.

²http://www4.stat.ncsu.edu/\,hdbondel/software.html

³relabeled for better clarity as ‘3 − the original label’.