Quantile regression allows one to estimate and conduct inference about the conditional quantile functions. It is somewhat analogous to ordinary least squares, which instead concerns the conditional mean.
Let be a random variable with cumulative distribution function . For , the -th quantile of is defined as
is called the quantile function. Like the distribution function, it provides a complete characterization of . The median, is a special case.
The loss function in the quantile regression framework is the so called “check function”
For some parametric function , the quantile regression estimate of minimizes the objective function
- Koenker, R., and G. W. Bassett (1978). Regression Quantiles. Econometrica 46, 33–50.
- Quantile Regression in WikiBooks:Statistics:Numerical Methods/Quantile Regression.