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Quantile regression

Quantile regression allows one to estimate and conduct inference about the conditional quantile functions. It is analogous to ordinary least squares? which concerns the conditional mean.

Quantiles

Let $Y$ be a random variable with cumulative distribution function $F(y)=\mathrm{Pr}(Y\le y)$. For $0\le \tau <1$, the $\tau$-th quantile of $Y$ is defined as

$Q(\tau )$ is called the quantile function. Like the distribution function, it provides a complete characterization of $Y$. The median, $Q(0.5)$ is a special case.

Regression

The loss function in the quantile regression framework is the so called “check function”

For some parametric function $q(x_i,\beta )$, the quantile regression estimate $\hat{\beta }$ of $\beta$ maximizes the objective function

References

• Koenker, R., and G. W. Bassett (1978). Regression Quantiles. Econometrica 46, 33–50.

External Links

• Quantile Regression in WikiBooks:Statistics:Numerical Methods/Quantile Regression.