Quantile Regression - Quantiles

Quantiles

Let be a real valued random variable with distribution function . The th quantile of Y is given by

where

Define the loss function as . A specific quantile can be found by minimizing the expected loss of with respect to :

This can be shown by setting the derivative of the expected loss function to 0 and letting be the solution of

This equation reduces to

and then to

Hence is th quantile of the random variable Y.