Open Access
August 2009 Local linear spatial quantile regression
Marc Hallin, Zudi Lu, Keming Yu
Bernoulli 15(3): 659-686 (August 2009). DOI: 10.3150/08-BEJ168

Abstract

Let $\{(Y_{\mathbf i}, \mathbf X_{\mathbf i }), \mathbf i ∈ℤ^N\}$ be a stationary real-valued $(d+1)$-dimensional spatial processes. Denote by $\mathbf x ↦q_p(\mathbf x ), p∈(0, 1), \mathbf x ∈ℝ^d$, the spatial quantile regression function of order $p$, characterized by $\mathrm{P}\{Y_{\mathbf i} ≤q_p(\mathbf x )|\mathbf X _{\mathbf i }=\mathbf x \}=p$. Assume that the process has been observed over an $N$-dimensional rectangular domain of the form $\mathcal{I}_{\mathbf{n}}:=\{\mathbf{i}=(i_{1},\ldots,i_{N})\in\mathbb{Z}^{N}\vert1\leq i_{k}\leq n_{k},k=1,\ldots,N\}$, with $\mathbf n =(n_1, …, n_N)∈ℤ^N$. We propose a local linear estimator of $q_p$. That estimator extends to random fields with unspecified and possibly highly complex spatial dependence structure, the quantile regression methods considered in the context of independent samples or time series. Under mild regularity assumptions, we obtain a Bahadur representation for the estimators of $q_p$ and its first-order derivatives, from which we establish consistency and asymptotic normality. The spatial process is assumed to satisfy general mixing conditions, generalizing classical time series mixing concepts. The size of the rectangular domain $\mathcal{I}_{\mathbf{n}}$ is allowed to tend to infinity at different rates depending on the direction in $ℤ^N$ (non-isotropic asymptotics). The method provides much richer information than the mean regression approach considered in most spatial modelling techniques.

Citation

Download Citation

Marc Hallin. Zudi Lu. Keming Yu. "Local linear spatial quantile regression." Bernoulli 15 (3) 659 - 686, August 2009. https://doi.org/10.3150/08-BEJ168

Information

Published: August 2009
First available in Project Euclid: 28 August 2009

zbMATH: 05815950
MathSciNet: MR2555194
Digital Object Identifier: 10.3150/08-BEJ168

Keywords: Bahadur representation , local linear estimation , Quantile regression , Random fields

Rights: Copyright © 2009 Bernoulli Society for Mathematical Statistics and Probability

Vol.15 • No. 3 • August 2009
Back to Top