Suppose $F : \mathbb{R} \to \mathbb{R}$ is a nondecreasing and right-continuous function. Define $G : [\inf F,\sup F] \to \overline{\mathbb{R}}$ by $G(p)=\inf \{ x : F(x) \geq p \}$, with the convention $\inf \emptyset = +\infty$. If $F$ is invertible then this is the inverse of $F$, but it makes sense even if $F$ is neither surjective nor injective. Is there a standard term for this $G$?
An application of this idea is to sampling random variables. Specifically, given $X$ distributed as $U(0,1)$ and a CDF $F$, $G(X)$ has the CDF $F$. If there is a standard term for $G$ only in this context, I would be happy to hear that as well.
Best Answer
It looks like the quantile function. It's domain is usually $(0, 1)$ and here are some simple properties:
$F^{- 1} (x) \leqslant t$ iff $x \leqslant F (t)$.
$F^{- 1}$ is increasing and left-continuous.
If $F$ is continuous, then $F \circ F^{- 1} = \text{id}$.