Quantile regression

One advantage of quantile regression relative to 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 this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be used to more comprehensively analyze the relationship between variables.{{cite book|title=Quantile Regression|url=https://archive.org/details/quantileregressi00koen_757|url-access=limited|last=Koenker|first=Roger|publisher=Cambridge University Press|year=2005|isbn=978-0-521-60827-5|pages=[https://archive.org/details/quantileregressi00koen_757/page/n151 146]–7}}

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.{{cite journal |first1=Brian S. |last1=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 }}

Another application of quantile regression is in the areas of growth charts, where percentile curves are commonly used to screen for abnormal growth.{{cite journal |last1=Wei |first1=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 |volume=25 |issue=8 |pages=1369–1382 |doi=10.1002/sim.2271 |pmid=16143984 |s2cid=7830193 }}{{cite journal |last1=Wei |first1=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 |arxiv=math/0702634 |s2cid=88516697 }}

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 Ruđer Josip Bošković, a Jesuit Catholic priest from Dubrovnik.{{rp|4}}{{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 }} 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.{{Cite book|title=Quantile Regression|url=https://archive.org/details/quantileregressi00koen_757|url-access=limited|last=Koenker|first=Roger|publisher=Cambridge University Press|year=2005|isbn=9780521845731|location=Cambridge|pages=[https://archive.org/details/quantileregressi00koen_757/page/n7 2]}} He finally produced the first geometric procedure for determining the equator of a rotating planet from three 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 in 1805 by fifty years.{{Cite book|title=Quantile Regression: Estimation and Simulation|last1=Furno|first1=Marilena|last2=Vistocco|first2=Domenico|publisher=John Wiley & Sons|year=2018|isbn=9781119975281|location=Hoboken, NJ|pages=xv}}

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 Edgeworth's plural median{{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}} - a geometric approach to median regression - and is recognized as the precursor of the simplex method. 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.

Quantile regression expresses the conditional quantiles of a dependent variable as a linear function of the explanatory variables. Crucial to the practicality of quantile regression is that the quantiles can be expressed as the solution of a minimization problem, as we will show in this section before discussing conditional quantiles in the next section.

{{Main|Quantile function}}

Let $Y$ be a real-valued random variable with cumulative distribution function $F\_\{Y\}(y)=P(Y\backslash leq\; y)$. The $\backslash tau$th quantile of Y is given by

:$q\_\{Y\}(\backslash tau)=F\_\{Y\}^\{-1\}(\backslash tau)=\backslash inf\backslash left\backslash \{\; y:F\_\{Y\}(y)\backslash geq\backslash tau\backslash right\backslash \}$

where $\backslash tau\backslash in(0,1).$

Define the loss function as , where $\backslash mathbb\{I\}$ is an indicator function.

A specific quantile can be found by minimizing the expected loss of $Y-u$ with respect to $u$:({{pp.|5|6}}):

:$q\_\{Y\}(\backslash tau)=\backslash underset\{u\}\{\backslash mbox\{arg\; min\}\}E(\backslash rho\_\{\backslash tau\}(Y-u))=\backslash underset\{u\}\{\backslash mbox\{arg\; min\}\}\backslash biggl\backslash \{(\backslash tau-1)\backslash int\_\{-\backslash infty\}^\{u\}(y-u)dF\_\{Y\}(y)+\backslash tau\backslash int\_\{u\}^\{\backslash infty\}(y-u)dF\_\{Y\}(y)\backslash biggr\backslash \}.$

This can be shown by computing the derivative of the expected loss with respect to $u$ via an application of the Leibniz integral rule, setting it to 0, and letting $q\_\{\backslash tau\}$ be the solution of

:$0=(1-\backslash tau)\backslash int\_\{-\backslash infty\}^\{q\_\{\backslash tau\}\}dF\_\{Y\}(y)-\backslash tau\backslash int\_\{q\_\{\backslash tau\}\}^\{\backslash infty\}dF\_\{Y\}(y).$

This equation reduces to

:$0=F\_\{Y\}(q\_\{\backslash tau\})-\backslash tau,$

and then to

:$F\_\{Y\}(q\_\{\backslash tau\})=\backslash tau.$

If the solution $q\_\{\backslash tau\}$ is not unique, then we have to take the smallest such solution to obtain

the $\backslash tau$th quantile of the random variable Y.

Let $Y$ be a discrete random variable that takes values $y\_i\; =\; i$ with $i\; =\; 1,2,\backslash dots,9$ with equal probabilities. The task is to find the median of Y, and hence the value $\backslash tau=0.5$ is chosen. Then the expected loss of $Y-u$ is

:

Since $\{0.5/9\}$ is a constant, it can be taken out of the expected loss function (this is only true if $\backslash tau=0.5$). Then, at u=3,

:$L(3)\; \backslash propto\backslash sum\_\{i=1\}^\{2\}$$-(i-3)$$+\backslash sum\_\{i=3\}^\{9\}$$(i-3)$$=[(2+1)+(0+1+2+...+6)]\; =24.$

Suppose that u is increased by 1 unit. Then the expected loss will be changed by $(3)-(6)=-3$ on changing u to 4. If, u=5, the expected loss is

:$L(5)\; \backslash propto\; \backslash sum\_\{i=1\}^\{4\}i+\backslash sum\_\{i=0\}^\{4\}i=20,$

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 $\{0.5/9\}$) for different values of u.

Consider $\backslash tau=0.5$ and let q be an initial guess for $q\_\{\backslash tau\}$. The expected loss evaluated at q is

:$L(q)=-0.5\backslash int\_\{-\backslash infty\}^\{q\}(y-q)dF\_\{Y\}(y)+0.5\backslash int\_\{q\}^\{\backslash infty\}(y-q)dF\_\{Y\}(y)\; .$

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

:$\backslash int\_\{-\backslash infty\}^\{q\}1dF\_\{Y\}(y)-\backslash int\_\{q\}^\{\backslash infty\}1dF\_\{Y\}(y)\; .$

The first term of the equation is $F\_\{Y\}(q)$ and second term of the equation is $1-F\_\{Y\}(q)$. Therefore, the change of expected loss function is negative if and only if , 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 $100\backslash tau$% of the points.

The $\backslash tau$ sample quantile can be obtained by using an importance sampling estimate and solving the following minimization problem

:$\backslash hat\{q\}\_\{\backslash tau\}=\backslash underset\{q\backslash in\; \backslash mathbb\{R\}\}\{\backslash mbox\{arg\; min\}\}\backslash sum\_\{i=1\}^\{n\}\backslash rho\_\{\backslash tau\}(y\_\{i\}-q)\; ,$

:

where the function $\backslash rho\_\{\backslash tau\}$ is the tilted absolute value function. The intuition is the same as for the population quantile.

The $\backslash tau$th conditional quantile of $Y$ given $X$ is the $\backslash tau$th quantile of the

Conditional probability distribution of $Y$ given $X$,

:$Q\_\{Y|X\}(\backslash tau)=\backslash inf\backslash left\backslash \{\; y:F\_\{Y|X\}(y)\backslash geq\backslash tau\backslash right\backslash \}$.

We use a capital $Q$ to denote the conditional quantile to indicate that it is a random variable.

In quantile regression for the $\backslash tau$th quantile we make the assumption that the $\backslash tau$th conditional quantile is given as a linear function of the explanatory variables:

:$Q\_\{Y|X\}(\backslash tau)=X\backslash beta\_\{\backslash tau\}$.

Given the distribution function of $Y$, $\backslash beta\_\{\backslash tau\}$ can be obtained by solving

:$\backslash beta\_\{\backslash tau\}=\backslash underset\{\backslash beta\backslash in\; \backslash mathbb\{R\}^\{k\}\}\{\backslash mbox\{arg\; min\}\}E(\backslash rho\_\{\backslash tau\}(Y-X\backslash beta)).$

Solving the sample analog gives the estimator of $\backslash beta$.

:$\backslash hat\{\backslash beta\_\{\backslash tau\}\}=\backslash underset\{\backslash beta\backslash in\; \backslash mathbb\{R\}^\{k\}\}\{\backslash mbox\{arg\; min\}\}\backslash sum\_\{i=1\}^\{n\}(\backslash rho\_\{\backslash tau\}(Y\_\{i\}-X\_\{i\}\backslash beta))\; .$

Note that when $\backslash tau\; =\; 0.5$, the loss function $\backslash rho\_\backslash tau$ is proportional to the absolute value function, and thus median regression is the same as

linear regression by least absolute deviations.

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 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 the minimization problem can be reformulated as a linear programming problem

:$\backslash underset\{\backslash beta,u^\{+\},u^\{-\}\backslash in\; \backslash mathbb\{R\}^\{k\}\backslash times\; \backslash mathbb\{R\}\_\{+\}^\{2n\}\}\{\backslash min\}\backslash left\backslash \{\; \backslash tau1\_\{n\}^\{\text{'}\}u^\{+\}+(1-\backslash tau)1\_\{n\}^\{\text{'}\}u^\{-\}|X\backslash beta+u^\{+\}-u^\{-\}=Y\backslash right\backslash \}\; ,$

where

:$u\_\{j\}^\{+\}=\backslash max(u\_\{j\},0)$ ,    $u\_\{j\}^\{-\}=-\backslash min(u\_\{j\},0).$

Simplex methods{{rp|181}} or interior point methods{{rp|190}} can be applied to solve the linear programming problem.

For $\backslash tau\backslash in(0,1)$, under some regularity conditions, $\backslash hat\{\backslash beta\}\_\{\backslash tau\}$ is asymptotically normal:

:$\backslash sqrt\{n\}(\backslash hat\{\backslash beta\}\_\{\backslash tau\}-\backslash beta\_\{\backslash tau\})\backslash overset\{d\}\{\backslash rightarrow\}N(0,\backslash tau(1-\backslash tau)D^\{-1\}\backslash Omega\_\{x\}D^\{-1\}),$

where

:$D=E(f\_\{Y\}(X\backslash beta)XX^\{\backslash prime\})$ and $\backslash Omega\_\{x\}=E(X^\{\backslash prime\}\; X)\; .$

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.{{cite journal |last1=Kocherginsky |first1=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 |s2cid=120598656 }}

See invariant estimator for background on invariance or see equivariance.

For any $a>0$ and $\backslash tau\backslash in[0,1]$

:$\backslash hat\{\backslash beta\}(\backslash tau;aY,X)=a\backslash hat\{\backslash beta\}(\backslash tau;Y,X),$

:$\backslash hat\{\backslash beta\}(\backslash tau;-aY,X)=-a\backslash hat\{\backslash beta\}(1-\backslash tau;Y,X).$

For any $\backslash gamma\backslash in\; R^\{k\}$ and $\backslash tau\backslash in[0,1]$

:$\backslash hat\{\backslash beta\}(\backslash tau;Y+X\backslash gamma,X)=\backslash hat\{\backslash beta\}(\backslash tau;Y,X)+\backslash gamma\; .$

Let $A$ be any $p\backslash times\; p$ nonsingular matrix and $\backslash tau\backslash in[0,1]$

:$\backslash hat\{\backslash beta\}(\backslash tau;Y,XA)=A^\{-1\}\backslash hat\{\backslash beta\}(\backslash tau;Y,X)\; .$

If $h$ is a nondecreasing function on $\backslash mathbb\{R\}$, the following invariance property applies:

:$h(Q\_\{Y|X\}(\backslash tau))\backslash equiv\; Q\_\{h(Y)|X\}(\backslash tau).$

Example (1):

If $W=\backslash exp(Y)$ and $Q\_\{Y|X\}(\backslash tau)=X\backslash beta\_\{\backslash tau\}$, then $Q\_\{W|X\}(\backslash tau)=\backslash exp(X\backslash beta\_\{\backslash tau\})$. The mean regression does not have the same property since $\backslash operatorname\{E\}\; (\backslash ln(Y))\backslash neq\; \backslash ln(\backslash operatorname\{E\}(Y)).$

The linear model $Q\_\{Y|X\}(\backslash tau)=X\backslash beta\_\{\backslash tau\}$ mis-specifies the true systematic relation $Q\_\{Y|X\}(\backslash tau)=f(X,\backslash tau)$ when $f(\backslash cdot,\backslash tau)$ is nonlinear. However, $Q\_\{Y|X\}(\backslash tau)=X\backslash beta\_\{\backslash tau\}$ minimizes a weighted distanced to $f(X,\backslash tau)$ among linear models.{{cite journal |last1=Angrist |first1=J. |last2=Chernozhukov |first2=V. | last3=Fernandez-Val |first3=I.| year=2006 |title=Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure |journal=Econometrica |volume=74 |issue=2 |pages=539–563 |doi=10.1111/j.1468-0262.2006.00671.x |url=http://papers.nber.org/papers/w10428.pdf }} Furthermore, the slope parameters $\backslash beta\_\{\backslash tau\}$ of the linear model can be interpreted as weighted averages of the derivatives $\backslash nabla\; f(X,\backslash tau)$ so that $\backslash beta\_\{\backslash tau\}$ can be used for causal inference.{{cite journal |last1=Kato |first1=R. |last2=Sasaki |first2=Y. | year=2017 |title=On Using Linear Quantile Regressions for Causal Inference |journal=Econometric Theory |volume=33 |issue=3 |pages=664–690 |doi=10.1017/S0266466616000177 |doi-access=free }} Specifically, the hypothesis $H\_0:\; \backslash nabla\; f(x,\backslash tau)=0$ for all $x$ implies the hypothesis $H\_0:\; \backslash beta\_\backslash tau=0$, which can be tested using the estimator $\backslash hat\{\backslash beta\_\{\backslash tau\}\}$ and its limit distribution.

The goodness of fit for quantile regression for the $\backslash tau$ quantile can be defined as:Roger Koenker & José A. F. Machado (1999) Goodness of Fit and Related Inference Processes for Quantile Regression, Journal of the American Statistical Association, 94:448, 1296-1310, DOI: 10.1080/01621459.1999.10473882

$R^2(\backslash tau)=1-\backslash frac\{\backslash hat\{V\}\_\backslash tau\}\{\backslash tilde\{V\}\_\backslash tau\},$

where $\backslash hat\{V\}\_\backslash tau$ is the sum of squares of the conditional quantile, while $\backslash tilde\{V\}\_\backslash tau$ is the sum of squares of the unconditional quantile.

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,{{cite journal |last1=Kozumi |first1=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 |s2cid=44015988 |url=https://www.b.kobe-u.ac.jp/papers_files/2009_02.pdf }} 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{{cite journal |last1=Yang |first1=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 |hdl=2027.42/135059 |s2cid=14947362 |hdl-access=free }} provided a posterior variance adjustment for valid inference. In addition,

Yang and He{{cite journal |last1=Yang |first1=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 |s2cid=88519086 }} showed that one can have asymptotically valid posterior inference if the working likelihood is chosen to be the empirical likelihood.

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 (a.k.a. the pinball loss{{cite journal | last1 = Steinwart | first1 = Ingo | last2 = Christmann | first2 = Andreas | title = Estimating conditional quantiles with the help of the pinball loss | journal = Bernoulli | volume = 17 | issue = 1 | pages = 211–225 | year = 2011 | publisher = Bernoulli Society for Mathematical Statistics and Probability | doi = 10.3150/10-BEJ267 | url = https://doi.org/10.3150/10-BEJ267| arxiv = 1102.2101 }}) 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,{{cite arXiv | last=Petneházi | first=Gábor | title=QCNN: Quantile Convolutional Neural Network | date=2019-08-21 | eprint=1908.07978 | class=cs.LG }}{{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 }} which is then referred to as nonparametric quantile regression.Nonparametric Quantile Regression: Non-Crossing Constraints and Conformal Prediction by Wenlu Tang, Guohao Shen, Yuanyuan Lin, Jian Huang, https://arxiv.org/pdf/2210.10161.pdf

Tree-based learning algorithms are also available for quantile regression (see, e.g., Quantile Regression Forests,{{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 }} as a simple generalization of Random Forests).

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{{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 |s2cid=120674851 }}

and Wang and Wang{{cite journal |last1=Wang |first1=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 |s2cid=34494316 }}

Example (2):

Let $Y^\{c\}=\backslash max(0,Y)$ and $Q\_\{Y|X\}=X\backslash beta\_\{\backslash tau\}$. Then $Q\_\{Y^\{c\}|X\}(\backslash tau)=\backslash max(0,X\backslash beta\_\{\backslash tau\})$. This is the censored quantile regression model: estimated values can be obtained without making any distributional assumptions, but at the cost of computational difficulty,{{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 }} some of which can be avoided by using a simple three step censored quantile regression procedure as an approximation.{{cite journal |first1=Victor |last1=Chernozhukov |first2=Han |last2=Hong |title=Three-Step Censored Quantile Regression and Extramarital Affairs |journal=J. Amer. Statist. Assoc. |volume=97 |issue=459 |pages=872–882 |year=2002 |doi=10.1198/016214502388618663 |s2cid=1410755 }}

For random censoring on the response variables, the censored quantile regression of Portnoy (2003) provides consistent estimates of all identifiable quantile functions based on reweighting each censored point appropriately.

Censored quantile regression has close links to survival analysis.

File:Km plot.jpgs for the survival probabilities $S(t)=1-F(t)$ of two patient groups as a function of time $t$, where $F(t)$ is the distribution function of the deaths. The $\backslash tau$ quantile of the deaths is $t\_\backslash tau\; =\; F^\{-1\}(\backslash tau)$, where $F^\{-1\}$ is the quantile function of the deaths. Censored quantile regression can be used to estimate these conditional quantiles individually, while survival analysis estimates the (conditional) survival function.]]

The quantile regression loss needs to be adapted in the presence of heteroscedastic errors in order to be efficient.Efficient Quantile Regression for Heteroscedastic

Models by, Yoonsuh Jung, Yoonkyung Lee, Steven N. MacEachern, https://www.tandfonline.com/doi/abs/10.1080/00949655.2014.967244?journalCode=gscs20

Numerous statistical software packages include implementations of quantile regression:

• Matlab function quantreg{{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}}
• gretl has the quantreg command.{{Cite web|title = Gretl Command Reference|url = http://ricardo.ecn.wfu.edu/pub//gretl/manual/en/gretl-ref.pdf|date = April 2017|access-date = 2017-04-22|archive-date = 2018-12-15|archive-url = https://web.archive.org/web/20181215121225/http://ricardo.ecn.wfu.edu/pub//gretl/manual/en/gretl-ref.pdf|url-status = dead}}
• R offers several packages that implement quantile regression, most notably quantreg by Roger Koenker,{{cite web |title=quantreg: Quantile Regression |work=R Project |date= 2018-12-18|url=https://cran.r-project.org/web/packages/quantreg/index.html }} but also gbm,{{cite web |title=gbm: Generalized Boosted Regression Models |work=R Project |date= 2019-01-14|url=https://cran.r-project.org/web/packages/gbm/ }} quantregForest,{{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 }} qrnn{{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 }} and qgam{{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 }}
• Python, via Scikit-garden{{cite web |title=Quantile Regression Forests |url=https://scikit-garden.github.io/examples/QuantileRegressionForests/ |website=Scikit-garden |access-date=3 January 2019}} and statsmodels{{cite web |title=Statsmodels: Quantile Regression |url=https://www.statsmodels.org/dev/examples/notebooks/generated/quantile_regression.html |website=Statsmodels |access-date=15 November 2019}}
• SAS through proc quantreg (ver. 9.2){{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 }} and proc quantselect (ver. 9.3).{{cite web |title=The QUANTSELECT Procedure |work=SAS Support |url=https://support.sas.com/rnd/app/stat/procedures/quantselect.html }}
• Stata, via the qreg command.{{cite web |title=qreg — Quantile regression |work=Stata Manual |url=https://www.stata.com/manuals15/rqreg.pdf }}{{cite book |first1=A. Colin |last1=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 |chapter-url=https://books.google.com/books?id=UkKQRAAACAAJ&pg=PA211 }}
• Vowpal Wabbit, via --loss_function quantile.{{Cite web|url=https://github.com/JohnLangford/vowpal_wabbit/wiki/Loss-functions|title=JohnLangford/vowpal_wabbit|website=GitHub|access-date=2016-07-09}}
• Mathematica package QuantileRegression.m{{cite web |title=QuantileRegression.m |url=https://github.com/antononcube/MathematicaForPrediction/blob/master/QuantileRegression.m |website=MathematicaForPrediction |access-date=3 January 2019}} hosted at the MathematicaForPrediction project at GitHub.
• Wolfram Language function QuantileRegression{{cite web |title=QuantileRegression |url=https://resources.wolframcloud.com/FunctionRepository/resources/QuantileRegression |website=Wolfram Function Repository |access-date=14 September 2022}} hosted at Wolfram Function Repository.

• Least-absolute-deviations regression
• Conformal prediction

{{wikibooks

|1= R Programming

|2= Quantile Regression

}}

• {{cite book |last1=Angrist |first1=Joshua D. |author-link=Joshua Angrist |last2=Pischke |first2=Jörn-Steffen |title=Mostly Harmless Econometrics: An Empiricist's Companion |publisher=Princeton University Press |year=2009 |isbn=978-0-691-12034-8 |chapter=Quantile Regression |pages=269–291 |chapter-url=https://books.google.com/books?id=ztXL21Xd8v8C&pg=PA269 }}
• {{cite book |last=Koenker |first=Roger |author-link=Roger Koenker |year=2005 |title=Quantile Regression |publisher=Cambridge University Press |isbn=978-0-521-60827-5 |url=https://books.google.com/books?id=hdkt7V4NXsgC }}

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Category:Regression analysis