I was wondering what would be the most efficient algorithm to solve quantile regression with a single predictor and no intercept? I tried a Brent line search, but unfortunately that's no faster than the interior point method implemented in quantreg (see below). So I was wondering if there would be potentially faster ways to do a quantreg model with a single predictor? Ideally I would also like to be able to use warm starts, which would exclude interior point methods...

EDIT: In this paper, "cqrReg: An R Package for Quantile and Composite Quantile Regression and Variable Selection", on page 14 I see a coordinate descent algorithm mentioned to solve quantile regression (implemented in R in https://github.com/cran/cqrReg/blob/master/R/QR.cd.R and https://github.com/cran/cqrReg/blob/master/src/QRCD.cpp; qrfit(x, y, tau, beta0, method="cd")). This might fit the bill, except that the current implementation always fits an intercept, which I don't want. Would anyone know if that algorithm could be adapted to also allow models to be fit without intercept, and what changes in the Rcpp code this would entail?

a=rnorm(n*p, mean = 10, sd = 1)
tau=0.01 # example tau, here to get underapproximation


# quantreg solution
microbenchmark(microbenchmark(rq.fit.br(x, y, tau = tau)$coefficients)) # quantreg, 149 µs, beta=0.84

# using Brent line search, using [0, least square estimate] search interval
# function giving log(quantile loss)
logquantloss = function(beta, x, y, tau) { r=y-beta*x   
                                           log(sum(r*(r>0) - (1-tau)*r)) } 
                       interval=c(0, mean(x*y)/mean(x*x)), # least square beta = maximum
                       x=x, y=y, tau=tau)$minimum) # 184 µs, beta=0.84
  • 2
    $\begingroup$ When all regressors are positive, clearly the solution is given by a suitable quantile of the $y_i/x_i,$ which is found with $O(n)$ effort. The generalization to data that straddle zero shouldn't be much harder. $\endgroup$
    – whuber
    Feb 21, 2019 at 21:53
  • $\begingroup$ Ha yes of course - stupid that I hadn't thought of that... microbenchmark(quantile(y/x,tau)) runs in 139 µs and indeed gives 0.844 in my example... Next question then would be if you would happen to know of the fastest Rcpp quantile() implementation in R that runs in O(n) time? $\endgroup$ Feb 21, 2019 at 22:55
  • $\begingroup$ I don't--but it's easy to code yourself. Emulate a Quicksort algorithm or use a heap. $\endgroup$
    – whuber
    Feb 22, 2019 at 12:24
  • $\begingroup$ Thanks - this Rcpp quantile function here, stackoverflow.com/questions/26786078/…, is already 4x faster than quantile() but will keep on looking. Quicksort is only O(n log n) efficient though isn't it, not O(n)? $\endgroup$ Feb 22, 2019 at 14:35
  • 1
    $\begingroup$ Thanks for the pointers - found what I need here: yiqun-dai.blogspot.com/2017/03/… and that runs in 6 µs as opposed to 139 µs for my example :-) Great! $\endgroup$ Feb 22, 2019 at 15:24

1 Answer 1


I just ran across this and thought I should try to clarify a couple of points. It it true that the the quantile regression solution to this "regression through the origin" problem is a quantile of the y_i/x_i values, but it is a weighted quantile with weights |x_i|. Floyd and Rivest's SELECT is somewhat quicker than Hoare's quickselect for ordinary sample quantiles, but an efficient adaptation for weighted quantiles is to my knowledge still open. I've written a note about this here: http://www.econ.uiuc.edu/~roger/research/vinaigrettes/kuantile.pdf

  • $\begingroup$ Many thanks for that! So if I get you right the solution of a zero-intercept quantile regression is a weighted quantile y_i/x_i with weights abs(x_i), right? Would this function search.r-project.org/CRAN/refmans/robsurvey/html/… be a reasonably efficient implementation for that (it's using quickselect)? I was also wondering what the solution would be of a univariate zero intercept weighted quantile regression with weights w_i? The weighted quantile of y_i/x_i with weights abs(x_i*w_i) or something like that? Any thoughts or references/pointers? $\endgroup$ May 16 at 9:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.