We already know that the estimator of quantile regression defined by LAD (Least Absolut Deviation), minimizes sigma |e_i|.

enter image description here

But I also found a formula that minimize with an integral, not a sigma

enter image description here

Actually I can't understand why there are 2 different formulas. In what case that use a sigma or integral ? Are the distributions effect these formulas ?

  • 2
    $\begingroup$ Where do the formulas come from, and what context are they presented in? What have they defined the symbols as in each case? $\endgroup$
    – Glen_b
    Jan 27, 2017 at 6:29
  • 3
    $\begingroup$ Conceptually, integration is a summation. Summing over a region and integrating over some region are conceptually similar. The top formula looks like it's estimating a linear function to give the quantile $q$. The bottom formula like it's estimating the value $q$ for quantile $\theta$ of a random variable with CDF $F_Y$. In the top, you're summing over your observations (basically, using the probability measure implicitly defined by your sample) and in the bottom, you're integrating over $dF$. $\endgroup$ Jan 27, 2017 at 7:07
  • $\begingroup$ See stats.stackexchange.com/questions/251600 for an explanation of the integral expression and stats.stackexchange.com/questions/73623 for the connection between the sums and the integrals. $\endgroup$
    – whuber
    Apr 17, 2018 at 13:02

1 Answer 1


The integral is used to define the (population) quantile of the random variable $Y$, while the summation is for the linear quantile regression, which aims at estimating the regression quantile coefficients $\beta_q$ using data $(y_i,x_i),i=1,\ldots,N$. Both employ the quantile check loss $\rho_{\theta}(u)=u(\theta-I(u<0))$. Indeed, the integral can be viewed as $E[\rho_{\theta}(Y-q)]$ and the summation is $\sum_{i=1}^N\rho_{\theta}(y_i-x_i'\beta_q)$. An analogy is the definition of the expectation vs. that of the least squares estimation. Specifically, $E(Y)$ can be viewed as the minimizer of the integral $\int_{-\infty}^{\infty}(y-m)^2\mathrm{d}F_Y(y)$ and the least squares estimator (of the coefficient vector) is obtained by minimizing $\sum_{i=1}^N(y_i-x_i'\beta_m)^2$.


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.