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.
Read more about this topic: Quantile Regression
Famous quotes containing the word intuition:
“Reason is sight. Instinct is touch. Intuition is smell.”
—Mason Cooley (b. 1927)
“When a scientist is ahead of his times, it is often through misunderstanding of current, rather than intuition of future truth. In science there is never any error so gross that it won’t one day, from some perspective, appear prophetic.”
—Jean Rostand (1894–1977)