Multiple-Output Quantile Regression

Authored by: Marc Hallin , Miroslav Šiman

Handbook of Quantile Regression

Print publication date:  October  2017
Online publication date:  October  2017

Print ISBN: 9781498725286
eBook ISBN: 9781315120256
Adobe ISBN:

10.1201/9781315120256-12

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Abstract

Quantile regression is about estimating the quantiles of some d-dimensional response Y $ \mathbf Y $ conditional on the values x ∈ R p $ \mathbf{x}\in \mathbb R ^p $ of some covariates X $ \mathbf{X} $ . The problem is well understood when d = 1 $ d=1 $ (single-output case, where Y is used instead of Y $ \mathbf Y $ ): for a (conditional) probability distribution P Y = P X = x Y $ \mathrm{P}^{ Y}=\mathrm P^{ Y}_{\mathbf{X}=\mathbf{x}} $ on R $ \mathbb R $ , with distribution function F = F X = x $ F=F_{\mathbf{X}=\mathbf{x}} $ , the (conditional on X = x $ \mathbf{X}=\mathbf{x} $ ) quantile of order τ $ \tau $ of Y $ \mathbf Y $ is q τ ( x ) : = inf { y : F ( y ) ≥ τ } , τ ∈ [ 0 , 1 ) . $$ q_{\tau } (\mathbf{x}):= \inf \{y \ : \ F(y) \ge \tau \},\quad \tau \in [0, 1). $$

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