Quantile Regression - Intuition

Intuition

Consider and let q be an initial guess for . The expected loss evaluated at q is

In order to miminize the expected loss, we move the value of q a little bit to see whether the expect loss will rise or fall. Suppose we increase q by 1 unit. Then the change of expected loss would be

The first term of the equation is and second term of the equation is . 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 % of the points.