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.
Read more about this topic: Quantile Regression