Convolution-type smoothed quantile regression


The conquer library performs fast and accurate convolution-type smoothed quantile regression (Fernandes, Guerre and Horta, 2021, He et al., 2022, Tan, Wang and Zhou, 2022 for low/high-dimensional estimation and bootstrap inference.

In the low-dimensional setting, efficient gradient-based methods are employed for fitting both a single model and a regression process over a quantile range. Normal-based and (multiplier) bootstrap confidence intervals for all slope coefficients are constructed. In high dimensions, the conquer methods complemented with 1-penalization and iteratively reweighted 1-penalization are used to fit sparse models.



We are adding more flexible penalties and inference methods into the package.

2022-02-12 (Version 1.2.2):

Remove the unnecessary dependent packge caret for a cleaner installation.

2021-10-24 (Version 1.2.1):

Major updates:

  1. Add a function conquer.process for conquer process over a quantile range.

  2. Add functions conquer.reg, for high-dimensional conquer with Lasso, SCAD and MCP penalties. The first function is called with a prescribed λ, and the second function calibrate λ via cross-validation. The candidates of λ can be user-specified, or automatically generated by simulating the pivotal quantity proposed in Belloni and Chernozhukov, 2011.

Minor updates:

  1. Add logistic kernel for all the functions.

  2. Modify initialization using asymmetric Huber regression.

  3. Default number of tightening iterations is now 3.

  4. Parameters for SCAD (default = 3.7) and MCP (default = 3) are added as arguments into the functions.


conquer is available on CRAN, and it can be installed into R environment:


Common errors or warnings

A collection of error / warning messages and their solutions:


There are 4 functions in this library:


Let us illustrate conquer by a simple example. For sample size n = 5000 and dimension p = 500, we generate data from a linear model yi = β0 + <xi, β> + εi, for i = 1, 2, … n. Here we set β0 = 1, β is a p-dimensional vector with every entry being 1, xi follows p-dimensional standard multivariate normal distribution (available in the library MASS), and εi is from t2 distribution.

n = 5000
p = 500
beta = rep(1, p + 1)
X = mvrnorm(n, rep(0, p), diag(p))
err = rt(n, 2)
Y = cbind(1, X) %*% beta + err

Then we run both quantile regression using package quantreg, with a Frisch-Newton approach after preprocessing (Portnoy and Koenker, 1997), and conquer (with Gaussian kernel) on the generated data. The quantile level τ is fixed to be 0.5.

tau = 0.5
start = Sys.time()
fit.qr = rq(Y ~ X, tau = tau, method = "pfn")
end = Sys.time()
time.qr = as.numeric(difftime(end, start, units = "secs"))
est.qr = norm(as.numeric(fit.qr$coefficients) - beta, "2")

start = Sys.time()
fit.conquer = conquer(X, Y, tau = tau)
end = Sys.time()
time.conquer = as.numeric(difftime(end, start, units = "secs"))
est.conquer = norm(fit.conquer$coeff - beta, "2")

It takes 7.4 seconds to run the standard quantile regression but only 0.2 seconds to run conquer. In the meanwhile, the estimation error is 0.5186 for quantile regression and 0.4864 for conquer. For readers’ reference, these runtimes are recorded on a Macbook Pro with 2.3 GHz 8-Core Intel Core i9 processor, and 16 GB 2667 MHz DDR4 memory. We refer to He et al., 2022 for a more extensive numerical study.

Getting help

Help on the functions can be accessed by typing ?, followed by function name at the R command prompt.

For example, ?conquer will present a detailed documentation with inputs, outputs and examples of the function conquer.



System requirements



Xuming He, Xiaoou Pan, Kean Ming Tan and Wen-Xin Zhou


Xiaoou Pan


Barzilai, J. and Borwein, J. M. (1988). Two-point step size gradient methods. IMA J. Numer. Anal. 8 141-148. Paper

Belloni, A. and Chernozhukov, V. (2011) 1-penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82-130. Paper

Fan, J., Liu, H., Sun, Q. and Zhang, T. (2018). I-LAMM for sparse learning: Simultaneous control of algorithmic complexity and statistical error. Ann. Statist. 46 814-841. Paper

Fernandes, M., Guerre, E. and Horta, E. (2021). Smoothing quantile regressions. J. Bus. Econ. Statist. 39 338-357, Paper

He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2022). Smoothed quantile regression with large-scale inference. J. Econometrics, to appear. Paper

Koenker, R. (2005). Quantile Regression. Cambridge Univ. Press, Cambridge. Book

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. Paper

Portnoy, S. and Koenker, R. (1997). The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators. Statist. Sci. 12 279–300. Paper

Tan, K. M., Wang, L. and Zhou, W.-X. (2022). High-dimensional quantile regression: convolution smoothing and concave regularization. J. Roy. Statist. Soc. Ser. B 84(1) 205-233. Paper