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Quantile regression is now a widely recognized useful alternative to the classical least-squares regression. It was introduced in the seminal paper of Koenker and Bassett (1978b). Given a response variable Y and a vector of covariates x $ \mathbf{x} $ , quantile regression estimates the effects of x $ \mathbf{x} $ on the conditional quantile of Y. Formally, the τ $ \tau $ th ( 0 < τ < 1 $ 0<\tau <1 $ ) conditional quantile of Y given x $ \mathbf{x} $ is defined as Q Y ( τ | x ) = inf { t : F Y | x ( t ) ≥ τ } $ Q_{Y}(\tau |\mathbf{x}) = \inf \{t:F_{Y|\mathbf{x}}(t) \ge \tau \} $ , where F Y | x $ F_{Y|\mathbf{x}} $ is the conditional cumulative distribution function of Y given x $ \mathbf{x} $ . An important special case of quantile regression is the least absolute deviation (LAD) regression (Koenker and Bassett, 1978a), which estimates the conditional median Q Y ( 0.5 | x ) $ Q_{Y}(0.5|\mathbf{x}) $ .
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