Quantile Function

probability-theory

Definition

Quantile Function

Given that cumulative distribution function $F$ is not guaranteed to be injective, the inverse $F^{-1}$ is not well-defineda

To circumvent this issue, a generalised inverse of $F$ can be defined as:

$$F^{-1}(p)=\inf \{x\mid F(x)\ge p\}\text{for}p\in (0,1)$$

This generalised inverse is known as the quantile function. Usually, if $F$ is invertible, then $F^{-1}$ is the “usual” inverse of $F$.

$$\forall p\in (0,1):x_p=F^{-1}(p)⟺F(x_p)=p$$

Then, the value $x_p$ is called p-quantile. Note that quantiles are not generally unique.