Quantile Function

Inverse function of CDF by reflecting at the diagonal. Uses minimum Quantile. We can use this function to directly calculate quantiles of a given CDF.

In R, this can for example be done using the qnorm commands or the quantile function for the empirical case.

• $Q_F^-$ is left continous
• $Q_F^+$ is right continous
• $F$ continous ←→ $Q_F^-,Q_F^+$ strictly monotonic
• $F$ strictly monotonic ←→ $Q_F^-,Q_F^+$ continous

The Inversion Lemma (Quantiles) basically proves that the Quantile Function is indeed the inverse function of the CDF.