View source for Quantile regression

{{cleanup|date=December 2010}}
{{Regression bar}}
'''Quantile regression''' is a type of [[regression analysis]] used in statistics and econometrics.  Whereas the [[method of least squares]] results in estimates of the conditional ''[[mean]]'' of the response variable given certain values of the predictor variables, quantile regression aims at estimating either the conditional [[median]] or other [[quantiles]] of the response variable. Essentially, quantile regression is the extension of linear regression and we use it when the conditions of linear regression are not applicable.

==Advantages and applications==

Quantile regression is desired if conditional quantile functions are of interest.  One advantage of quantile regression, relative to the ordinary least squares regression, is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond that.  Different measures of [[central tendency]] and [[statistical dispersion]] can be useful to obtain a more comprehensive analysis of the relationship between variables.<ref name="Koenker2005">{{cite book|title=Quantile Regression|last=Koenker|first=Roger|publisher=Cambridge University Press|year=2005|isbn=978-0-521-60827-5|location=|pages=146–7|ref=harv}}</ref>

In [[ecology]], quantile regression has been proposed and used as a way to discover more useful predictive relationships between variables in cases where there is no relationship or only a weak relationship between the means of such variables.  The need for and success of quantile regression in ecology has been attributed to the [[complexity]] of interactions between different factors leading to [[data]] with unequal variation of one variable for different ranges of another variable.<ref>{{cite journal |first=Brian S. |last=Cade |first2=Barry R. |last2=Noon |year=2003 |url=http://www.econ.uiuc.edu/~roger/research/rq/QReco.pdf |title=A gentle introduction to quantile regression for ecologists |journal=Frontiers in Ecology and the Environment |volume=1 |issue=8 |pages=412–420 |doi= 10.2307/3868138|jstor=3868138 }}</ref>

Another application of quantile regression is in the areas of growth charts, where percentile curves are commonly used to screen for abnormal growth.<ref>{{cite journal |last=Wei |first=Y. |last2=Pere |first2=A. |last3=Koenker |first3=R. |last4=He |first4=X. |year=2006 |title=Quantile Regression Methods for Reference Growth Charts |journal=[[Statistics in Medicine (journal)|Statistics in Medicine]] |volume=25 |issue=8 |pages=1369–1382 |doi=10.1002/sim.2271 |pmid=16143984 |ref=harv }}</ref><ref>{{cite journal |last=Wei |first=Y. |last2=He |first2=X.|author2-link=Xuming He |year=2006 |title=Conditional Growth Charts (with discussions) |journal=[[Annals of Statistics]] |volume=34 |issue=5 |pages=2069–2097 and 2126–2131 |doi=10.1214/009053606000000623 |ref=harv |arxiv=math/0702634 }}</ref>

==Mathematics==

The mathematical forms arising from quantile regression are distinct from those arising in the [[method of least squares]].  The method of least squares leads to a consideration of problems in an [[inner product space]], involving [[projection (mathematics)|projection]] onto subspaces, and thus the problem of minimizing the squared errors can be reduced to a problem in [[numerical linear algebra]].  Quantile regression does not have this structure, and instead leads to problems in [[linear programming]] that can be solved by the [[simplex method]].

==History==
The idea of estimating a median regression slope, a major theorem about minimizing sum of the absolute deviances and a geometrical algorithm for constructing median regression was proposed in 1760 by [[Roger Joseph Boscovich|Ruđer Josip Bošković]], a [[Society of Jesus|Jesuit Catholic]] priest from Dubrovnik.<ref name="Koenker2005" />{{rp|4}}<ref>{{cite journal |last=Stigler |first=S. |year=1984 |title=Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation |journal=Biometrika |volume=71 |issue=3 |pages=615–620 |doi=10.1093/biomet/71.3.615 }}</ref> He was interested in the ellipticity of the earth, building on Isaac Newton's suggestion that its rotation could cause it to bulge at the [[equator]] with a corresponding flattening at the poles.<ref>{{Cite book|title=Quantile Regression|last=Koenker|first=Roger|publisher=Cambridge University Press|year=2005|isbn=9780521845731|location=Cambridge|pages=2}}</ref> He finally produced the first geometric procedure for determining the [[equator]] of a rotating [[planet]] from three [[Observation|observations]] of a surface feature. More importantly for quantile regression, he was able to develop the first evidence of the least absolute criterion and preceded the least squares introduced by [[Legendre polynomials|Legendre]] in 1805 by fifty years.<ref name=":0">{{Cite book|title=Quantile Regression: Estimation and Simulation|last=Furno|first=Marilena|last2=Vistocco|first2=Domenico|publisher=John Wiley & Sons|year=2018|isbn=9781119975281|location=Hoboken, NJ|pages=xv}}</ref> 

Other thinkers began building upon Bošković's idea such as [[Pierre-Simon Laplace]], who developed the so-called "methode de situation." This led to [[Francis Ysidro Edgeworth|Francis Edgeworth]]'s plural median<ref>{{Cite web|url=http://www.econ.uiuc.edu/~roger/research/galton/Galton.pdf|title=Galton, Edgeworth, Frisch, and prospects for quantile regression in economics|last=Koenker|first=Roger|date=August 1998|website=UIUC.edu|access-date=August 22, 2018}}</ref> - a geometric approach to median regression - and is recognized as the precursor of the [[Simplex algorithm|simplex method]].<ref name=":0" /> The works of Bošković, Laplace, and Edgeworth were recognized as a prelude to [[Roger Koenker]]'s contributions to quantile regression.

Median regression computations for larger data sets are quite tedious compared to the least squares method, for which reason it has historically generated a lack of popularity among statisticians, until the widespread adoption of computers in the latter part of the 20th century.

==Quantiles==
Let <math>Y</math> be a real valued random variable with [[cumulative distribution function]] <math>F_{Y}(y)=P(Y\leq y)</math>. The <math>\tau</math>th quantile of Y is given by
:<math>Q_{Y}(\tau)=F_{Y}^{-1}(\tau)=\inf\left\{ y:F_{Y}(y)\geq\tau\right\}</math>
where <math>\tau\in(0,1).</math>

Define the [[loss function]] as <math>\rho_{\tau}(y)=y(\tau-\mathbb{I}_{(y<0)})</math>, where <math>\mathbb{I}</math> is an [[indicator function]]. A specific quantile can be found by minimizing the expected loss of <math>Y-u</math> with respect to <math>u</math>:<ref name="Koenker2005" />{{rp|5–6}}
:<math>\underset{u}{\min}E(\rho_{\tau}(Y-u))=\underset{u}{\min}\left\{(\tau-1)\int_{-\infty}^{u}(y-u)dF_{Y}(y)+\tau\int_{u}^{\infty}(y-u)dF_{Y}(y)\right\}.</math>

This can be shown by setting the derivative of the expected loss function to 0 and letting <math>q_{\tau}</math> be the solution of
:<math>0=(1-\tau)\int_{-\infty}^{q_{\tau}}dF_{Y}(y)-\tau\int_{q_{\tau}}^{\infty}dF_{Y}(y).</math>
This equation reduces to
:<math>0=F_{Y}(q_{\tau})-\tau,</math>
and then to
:<math>F_{Y}(q_{\tau})=\tau.</math>

Hence <math>q_{\tau}</math> is <math>\tau</math>th quantile of the random variable Y.

==Example==
Let <math>Y</math> be a discrete random variable that takes values 1,2,..,9 with equal probabilities. The task is to find the median of Y, and hence the value <math>\tau=0.5</math> is chosen. The expected loss, {{nowrap|1=''L''(''u'')}}, is
:<math>L(u)=\frac{(\tau-1)}{9}\sum_{y_{i}<u}(y_{i}-u)+\frac{\tau}{9}\sum_{y_{i}\geq u}(y_{i}-u)=\frac{0.5}{9}\left(-\sum_{y_{i}<u}(y_{i}-u)+\sum_{y_{i}\geq u}(y_{i}-u)\right) .</math>
Since <math>{0.5/9}</math> is a constant, it can be taken out of the expected loss function (this is only true if <math>\tau=0.5</math>). Then, at ''u''=3,
:<math>L(3) \propto\sum_{i=1}^{2}-(i-3)+\sum_{i=3}^{9}(i-3) =[(2+1)+(0+1+2+...+6)] =24.</math>

Suppose that ''u'' is increased by 1 unit. Then the expected loss will be changed by <math>(3)-(6)=-3</math> on changing ''u'' to 4. If,  ''u''=5, the expected loss is
:<math>L(5) \propto \sum_{i=1}^{4}i+\sum_{i=0}^{4}i=20,</math>
and any change in ''u'' will increase the expected loss. Thus ''u''=5 is the median. The Table below shows the expected loss (divided by <math>{0.5/9}</math>) for different values of ''u''.

{| class="wikitable"
|-
| ''u'' || 1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
| Expected loss||36|| 29|| 24|| 21|| 20|| 21|| 24|| 29 || 36
|}

==Intuition==
Consider <math>\tau=0.5</math> and let ''q'' be an initial guess for <math>q_{\tau}</math>. The expected loss evaluated at ''q'' is
:<math>L(q)=-0.5\int_{-\infty}^{q}(y-q)dF_{Y}(y)+0.5\int_{q}^{\infty}(y-q)dF_{Y}(y) .</math>
In order to minimize the expected loss, we move the value of ''q'' a little bit to see whether the expected loss will rise or fall.
Suppose we increase ''q'' by 1 unit. Then the change of expected loss would be
:<math>\int_{-\infty}^{q}1dF_{Y}(y)-\int_{q}^{\infty}1dF_{Y}(y) .</math>

The first term of the equation is <math>F_{Y}(q)</math> and second term of the equation is <math>1-F_{Y}(q)</math>. Therefore, the change of expected loss function is negative if and only if <math>F_{Y}(q)<0.5</math>, that is if and only if ''q'' is smaller than the median. Similarly, if we reduce ''q'' by 1 unit, the change of expected loss function is negative if and only if ''q'' is larger than the median.

In order to minimize the expected loss function, we would increase (decrease) ''L''(''q'') if ''q'' is smaller (larger) than the median, until ''q'' reaches the median. The idea behind the minimization is to count the number of points (weighted with the density) that are larger or smaller than ''q'' and then move ''q'' to a point where ''q'' is larger than <math>100\tau</math>% of the points.

==Sample quantile==
The <math>\tau</math> sample quantile can be obtained by solving the following minimization problem

:<math>\hat{q}_{\tau}=\underset{q\in \mathbb{R}}{\mbox{arg min}}\sum_{i=1}^{n}\rho_{\tau}(y_{i}-q) ,</math>
:<math>=\underset{q\in \mathbb{R}}{\mbox{arg min}}  \left[(\tau - 1)\sum_{y_{i}<q}(y_{i}-q)+\tau\sum_{y_{i}\geq q}(y_{i}-q) \right]</math>, where the function <math>\rho_{\tau}</math> is the tilted absolute value function. The intuition is the same as for the population quantile.

==Conditional quantile and quantile regression==

Suppose the <math>\tau</math>th conditional quantile function is <math> Q_{Y|X}(\tau)=X\beta_{\tau}</math>. Given the distribution function of <math>Y</math>,  <math>\beta_{\tau}</math> can be obtained by solving
:<math>\beta_{\tau}=\underset{\beta\in \mathbb{R}^{k}}{\mbox{arg min}}E(\rho_{\tau}(Y-X\beta)).</math>

Solving the sample analog gives the estimator of <math>\beta</math>.
:<math>\hat{\beta_{\tau}}=\underset{\beta\in \mathbb{R}^{k}}{\mbox{arg min}}\sum_{i=1}^{n}(\rho_{\tau}(Y_{i}-X_{i}\beta)) .</math>

==Computation==
The minimization problem can be reformulated as a [[linear programming]] problem
:<math>\underset{\beta,u^{+},u^{-}\in \mathbb{R}^{k}\times \mathbb{R}_{+}^{2n}}{\min}\left\{ \tau1_{n}^{'}u^{+}+(1-\tau)1_{n}^{'}u^{-}|X\beta+u^{+}-u^{-}=Y\right\} ,</math>

where
:<math>u_{j}^{+}=\max(u_{j},0)</math> ,&nbsp;&nbsp;&nbsp; <math>u_{j}^{-}=-\min(u_{j},0).</math>

[[Simplex algorithm|Simplex methods]]<ref name="Koenker2005" />{{rp|181}}  or [[interior point method]]s<ref name="Koenker2005" />{{rp|190}} can be applied to solve the linear programming problem.

==Asymptotic properties==

For <math>\tau\in(0,1)</math>, under some regularity conditions, <math>\hat{\beta}_{\tau}</math> is [[Asymptotic normality|asymptotically normal]]:
:<math>\sqrt{n}(\hat{\beta}_{\tau}-\beta_{\tau})\overset{d}{\rightarrow}N(0,\tau(1-\tau)D^{-1}\Omega_{x}D^{-1}),</math>
where
:<math>D=E(f_{Y}(X\beta)XX^{\prime})</math> and <math>\Omega_{x}=E(X^{\prime} X) .</math>

Direct estimation of the asymptotic variance-covariance matrix is not always satisfactory. Inference for quantile regression parameters can be made with the regression rank-score tests or with the bootstrap methods.<ref>{{cite journal |last=Kocherginsky |first=M. |last2=He |first2=X. |last3=Mu |first3=Y. |year=2005 |title=Practical Confidence Intervals for Regression Quantiles |journal=[[Journal of Computational and Graphical Statistics]] |volume=14 |issue=1 |pages=41–55 |doi=10.1198/106186005X27563 }}</ref>

==Equivariance==

See [[invariant estimator]] for background on invariance or see [[equivariance]].

===Scale equivariance===

For any <math>a>0</math> and <math>\tau\in[0,1]</math>
:<math>\hat{\beta}(\tau;aY,X)=a\hat{\beta}(\tau;Y,X),</math>
:<math>\hat{\beta}(\tau;-aY,X)=-a\hat{\beta}(1-\tau;Y,X).</math>

===Shift equivariance===
For any <math>\gamma\in R^{k}</math> and <math>\tau\in[0,1]</math>
:<math>\hat{\beta}(\tau;Y+X\gamma,X)=\hat{\beta}(\tau;Y,X)+\gamma .</math>

===Equivariance to reparameterization of design===
Let <math>A</math> be any <math>p\times p</math> nonsingular matrix and <math>\tau\in[0,1] </math>
:<math>\hat{\beta}(\tau;Y,XA)=A^{-1}\hat{\beta}(\tau;Y,X) .</math>

===Invariance to monotone transformations===
If <math>h</math> is a nondecreasing function on '''R'', the following [[Invariant estimator|invariance]] property applies:

:<math>h(Q_{Y|X}(\tau))\equiv Q_{h(Y)|X}(\tau).</math>

Example (1):

If <math>W=\exp(Y)</math> and <math>Q_{Y|X}(\tau)=X\beta_{\tau}</math>, then <math>Q_{W|X}(\tau)=\exp(X\beta_{\tau})</math>. The mean regression does not have the same property since <math>\operatorname{E} (\ln(Y))\neq \ln(\operatorname{E}(Y)).</math>

==Bayesian methods for quantile regression==

Because quantile regression does not normally assume a parametric likelihood for the conditional distributions of Y|X, the Bayesian methods work with a working likelihood. A convenient choice is the asymmetric Laplacian likelihood,<ref>{{cite journal |last=Kozumi |first=H. |last2=Kobayashi |first2=G. |year=2011 |title=Gibbs sampling methods for Bayesian quantile regression |journal=[[Journal of Statistical Computation and Simulation]] |volume=81 |issue=11 |pages=1565–1578 |doi=10.1080/00949655.2010.496117 |url=https://www.b.kobe-u.ac.jp/papers_files/2009_02.pdf }}</ref> because the mode of the resulting posterior under a flat prior is the usual quantile regression estimates. The posterior inference, however, must be interpreted with care.  Yang, Wang and He<ref>{{cite journal |last=Yang |first=Y. |last2=Wang |first2=H.X. | last3=He |first3=X.| year=2016 |title=Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood |journal=[[International Statistical Review]] |volume=84 |issue=3 |pages=327–344 |doi=10.1111/insr.12114 }}</ref> provided a posterior variance adjustment for valid inference. In addition,
Yang and He<ref>{{cite journal |last=Yang |first=Y. |last2=He |first2=X. |year=2010 |title=Bayesian empirical likelihood for quantile regression |journal=[[Annals of Statistics]] |volume=40 |issue=2 |pages=1102–1131 |doi=10.1214/12-AOS1005 |arxiv=1207.5378 }}</ref> showed that one can have asymptotically valid posterior inference if the working likelihood is chosen to be the empirical likelihood.

==Machine learning methods for quantile regression==
Beyond simple linear regression, there are several machine learning methods that can be extended to quantile regression. A switch from the squared error to the tilted absolute value loss function allows gradient descent based learning algorithms to learn a specified quantile instead of the mean. It means that we can apply all [[neural network]] and [[deep learning]] algorithms to quantile regression.<ref>{{cite arXiv | last=Petneházi | first=Gábor | title=QCNN: Quantile Convolutional Neural Network | date=2019-08-21 | eprint=1908.07978 | class=cs.LG }}</ref><ref>{{cite arXiv | last1=Rodrigues | first1=Filipe | last2=Pereira | first2=Francisco C. | title=Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems | date=2018-08-27 | eprint=1808.08798 | class=stat }}</ref> Tree-based learning algorithms are also available for quantile regression (see, e.g., Quantile Regression Forests<ref>{{cite journal | last=Meinshausen | first=Nicolai | title=Quantile Regression Forests | url=http://www.jmlr.org/papers/volume7/meinshausen06a/meinshausen06a.pdf | journal=Journal of Machine Learning Research | volume=7 | issue=6 | pages=983-999 | year=2006 }}</ref>, as a simple generalization of [[Random forest|Random Forests]]).

==Censored quantile regression==

If the response variable is subject to censoring, the conditional mean is not identifiable without additional distributional assumptions, but the conditional quantile is often identifiable. For recent work on censored quantile regression, see: Portnoy<ref name="Portnoy2003">{{cite journal |last=Portnoy |first=S. L. |year=2003 |title=Censored Regression Quantiles |journal=[[Journal of the American Statistical Association]] |volume=98 |issue=464 |pages=1001–1012 |doi=10.1198/016214503000000954 }}</ref>
and Wang and Wang<ref>{{cite journal |last=Wang |first=H. |author1-link= Huixia Judy Wang |last2=Wang |first2=L. |year=2009 |title=Locally Weighted Censored Quantile Regression |journal=[[Journal of the American Statistical Association]] |volume=104 |issue=487 |pages=1117–1128 |doi=10.1198/jasa.2009.tm08230 |citeseerx=10.1.1.504.796 }}</ref>

Example (2):

Let <math>Y^{c}=\max(0,Y)</math> and <math>Q_{Y|X}=X\beta_{\tau}</math>. Then <math>Q_{Y^{c}|X}(\tau)=\max(0,X\beta_{\tau})</math>. This is the censored quantile regression model: estimated values can be obtained without making any distributional assumptions, but at the cost of computational difficulty,<ref>{{cite journal |first=James L. |last=Powell |year=1986 |title=Censored Regression Quantiles |journal=[[Journal of Econometrics]] |volume=32 |issue=1 |pages=143–155 |doi=10.1016/0304-4076(86)90016-3 }}</ref> some of which can be avoided by using a simple three step censored quantile regression procedure as an approximation.<ref>{{cite journal |first=Victor |last=Chernozhukov |first2=Han |last2=Hong |title=Three-Step Censored Quantile Regression and Extramarital Affairs |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=97 |issue=459 |pages=872–882 |year=2002 |doi=10.1198/016214502388618663 }}</ref>

For random censoring on the response variables, the censored quantile regression of Portnoy (2003)<ref name="Portnoy2003" /> provides consistent estimates of all identifiable quantile functions based on reweighting each censored point appropriately.

==Implementations==
Numerous statistical software packages include implementations of quantile regression:
* [[MATLAB|Matlab]] function <code>quantreg</code><ref>{{Cite web|title = quantreg(x,y,tau,order,Nboot) - File Exchange - MATLAB Central|url = http://www.mathworks.com/matlabcentral/fileexchange/32115-quantreg-x-y-tau-order-nboot-|website = www.mathworks.com|access-date = 2016-02-01}}</ref>
* [[Eviews]], since version 6.{{citation needed|date=August 2015}}
* [[gretl]] has the <code>quantreg</code> command.<ref>{{Cite web|title = Gretl Command Reference|url = http://ricardo.ecn.wfu.edu/pub//gretl/manual/en/gretl-ref.pdf|date=April 2017}}</ref>
* [[R (programming language)|R]] offers several packages that implement quantile regression, most notably <code>quantreg</code> by [[Roger Koenker]],<ref>{{cite web |title=quantreg: Quantile Regression |work=R Project |date= 2018-12-18|url=https://cran.r-project.org/web/packages/quantreg/index.html }}</ref> but also <code>gbm</code>,<ref>{{cite web |title=gbm: Generalized Boosted Regression Models |work=R Project |date= 2019-01-14|url=https://cran.r-project.org/web/packages/gbm/ }}</ref> <code>quantregForest</code><ref>{{cite web |title=quantregForest: Quantile Regression Forests |work=R Project |date= 2017-12-19|url=https://cran.r-project.org/web/packages/quantregForest/index.html }}</ref>, <code>qrnn</code><ref>{{cite web |title=qrnn: Quantile Regression Neural Networks |work=R Project |date= 2018-06-26|url=https://cran.r-project.org/web/packages/qrnn/index.html }}</ref> and <code>qgam</code><ref>{{cite web |title=qgam: Smooth Additive Quantile Regression Models |work=R Project |date= 2019-05-23|url=https://cran.r-project.org/web/packages/qgam/index.html }}</ref>
* [[Python (programming language)|Python]], via <code>Scikit-garden</code><ref>{{cite web |title=Quantile Regression Forests |url=https://scikit-garden.github.io/examples/QuantileRegressionForests/ |website=Scikit-garden |accessdate=3 January 2019}}</ref> and <code>statsmodels</code><ref>{{cite web |title=Statsmodels: Quantile Regression |url=https://www.statsmodels.org/dev/examples/notebooks/generated/quantile_regression.html |website=Statsmodels |accessdate=15 November 2019}}</ref>
* [[SAS System|SAS]] through <code>proc quantreg</code> (ver. 9.2) and <code>proc quantselect</code> (ver. 9.3).<ref>{{cite web |title=An Introduction to Quantile Regression and the QUANTREG Procedure |work=SAS Support |url=http://www2.sas.com/proceedings/sugi30/213-30.pdf }}</ref>
* [[Stata]], via the <code>qreg</code> command.<ref>{{cite web |title=qreg — Quantile regression |work=Stata Manual |url=https://www.stata.com/manuals15/rqreg.pdf }}</ref><ref>{{cite book |first=A. Colin |last=Cameron |first2=Pravin K. |last2=Trivedi |title=Microeconometrics Using Stata |location=College Station |publisher=Stata Press |edition=Revised |year=2010 |isbn=978-1-59718-073-3 |chapter=Quantile Regression |pages=211–234 |chapterurl=https://books.google.com/books?id=UkKQRAAACAAJ&pg=PA211 }}</ref>
* [[Vowpal Wabbit]], via <code>--loss_function quantile</code>.<ref>{{Cite web|url=https://github.com/JohnLangford/vowpal_wabbit/wiki/Loss-functions|title=JohnLangford/vowpal_wabbit|website=GitHub|access-date=2016-07-09}}</ref>
* [[Statsmodels]] package for Python, via <code>QuantReg</code><ref>{{cite web |title=Quantile regression |url=http://statsmodels.sourceforge.net/devel/examples/notebooks/generated/quantile_regression.html |website=statsmodels.org |accessdate=3 January 2019}}</ref>
* [[Mathematica]] package <code>QuantileRegression.m</code><ref>{{cite web |title=QuantileRegression.m |url=https://github.com/antononcube/MathematicaForPrediction/blob/master/QuantileRegression.m |website=MathematicaForPrediction |accessdate=3 January 2019}}</ref> hosted at the MathematicaForPrediction project at GitHub.

==References==
{{reflist|30em}}

==Further reading==
{{wikibooks
 |1= R Programming
 |2= Quantile Regression
}}
* {{cite book |last=Angrist |first=Joshua D. |authorlink=Joshua Angrist |last2=Pischke |first2=Jörn-Steffen |title=Mostly Harmless Econometrics: An Empiricist's Companion |location= |publisher=Princeton University Press |year=2009 |isbn=978-0-691-12034-8 |chapter=Quantile Regression |pages=269–291 |chapterurl=https://books.google.com/books?id=ztXL21Xd8v8C&pg=PA269 }}
* {{cite book |last=Koenker |first=Roger |authorlink=Roger Koenker |year=2005 |title=Quantile Regression |publisher=Cambridge University Press |isbn=978-0-521-60827-5 |ref=harv |url=https://books.google.com/books?id=hdkt7V4NXsgC }}

{{DEFAULTSORT:Quantile Regression}}
[[Category:Regression analysis]]