Quantile regression

A method of estimating the parameters of a linear regression model is called quantile regression . In contrast to least squares estimation , which estimates the expected value of the target variable , the quantile regression is suitable for estimating its quantiles . The quantile regression is thus a possibility, by considering other properties of the target variable distribution, to give up the focus on the expected value of the target variable, which is based on the classic linear model . The median regression is a special case of quantile regression.

Optimization problem

Pinball loss function with . For is the error , for it is . ${\displaystyle \tau =0{,}9}$ ${\displaystyle \varepsilon <0}$ ${\displaystyle -0{,}1\varepsilon }$ ${\displaystyle \varepsilon \geq 0}$ ${\displaystyle 0{,}9\varepsilon }$

Be a real random variable with distribution function , then the equivalent - quantile of : ${\displaystyle Y}$ ${\displaystyle F_{Y}(y)=P(Y\leq y)}$ ${\displaystyle \tau }$ ${\displaystyle Y}$

${\displaystyle Q_{Y}(\tau )=F_{Y}^{-1}(\tau )=\inf \left\{y:F_{Y}(y)\geq \tau \right\}}$

With ${\displaystyle \tau \in (0,1).}$

Be with observed pairs of independent variables and associated dependent variable . The regression model is described as The optimal regression parameters can be determined by the following minimization: ${\displaystyle (\mathbf {x} _{i},y_{i})}$ ${\displaystyle i\in \{1,\dots ,n\}}$ ${\displaystyle \mathbf {x} _{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle y_{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i}}$

${\displaystyle {\hat {\boldsymbol {\beta }}}_{\tau }=\arg \min _{\beta _{\tau }}\sum _{i=1}^{n}w_{\tau }(y_{i},\eta _{i,\tau })|y_{i}-\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{\tau }|}$.

Here corresponds to the linear predictor . The loss function corresponds to the inclined absolute deviation: ${\displaystyle \eta _{i,\tau }=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{\tau }}$

${\displaystyle w_{\tau }(y_{i},\eta _{i,\tau })={\begin{cases}1-\tau &{\textrm {falls}}\quad y_{i}<\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{\tau }\\\tau &{\textrm {falls}}\quad y_{i}\geq \mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}_{\tau },\end{cases}}}$

where is the indicator function . Because of its appearance, the loss function is also called pinball loss . ${\displaystyle \mathbb {I} }$

The optimization problem can be solved with methods of linear programming , for example with the simplex method .

literature

• David J. Petersen et al .: Perspectives on a Plural Economy. Springer Vieweg . Springer Fachmedien, Wiesbaden 2019, ISBN 978-3-658-16144-6 , pp. 238-240.

Individual evidence

1. ^ David J. Petersen et al .: Perspectives of a Plural Economy. Springer Vieweg . Springer Fachmedien, Wiesbaden 2019, ISBN 978-3-658-16144-6 , p. 238.
2. ↑ Roger Koenker, Gilbert Basset Jr .: regression quantiles . In: Econometrica: journal of the Econometric Society . 1978, p. 33-50 .
3. ↑ Roger Koenker, Kevin F. Hallock: regression quantiles . In: Journal of economic perspectives . tape 15 , no. 4 , 2001, p. 143-156 .
4. ^ Ingo Steinwart, Andreas Christmann: Estimating conditional quantiles with the help of the pinball loss . In: Bernoulli . tape 17 , no. 1 , February 2011, ISSN  1350-7265 , p. 211–225 , doi : 10.3150 / 10-BEJ267 , arxiv : 1102.2101 ( projecteuclid.org [accessed July 11, 2020]).