[Math] Properties of inverse distribution function – continuity

probabilityprobability distributionsprobability theory

I have some questions about inverse distribution functions. Let $F : \mathbb R \to [0,1]$ be a distribution function and define $F^{-1} : [0,1] \to \overline{\mathbb R}$ by $F^{-1}(y) := \inf\{x \in \mathbb R; F(x) \ge y\}$ with the convention that $\inf \emptyset := +\infty$. Futhermore, define $F^{-1+} : [0, 1] \to \overline{\mathbb R}$ by $F^{-1+}(y) := \sup\{x \in \mathbb R; F(x) \le y\}$, where $\sup \emptyset := -\infty$.

The function $F^{-1}$ is the usual quantile function. I managed to prove the following properties:

  1. $F^{-1}$ and $F^{-1+}$ are non-decreasing.
  2. If $F^{-1} \in (-\infty, \infty)$, $F^{-1}$ is left-continuous at $y$ and admit a limit from the right at $y$.
  3. For $y \in \mathbb R$, set $A_y := \{x \in \mathbb R; F(x) = y\}$. Then, $(F^{-1}(y-), F^{-1}(y+)) \subseteq A_y$.
  4. Suppose that $F$ is strictly increasing. Then, $F^{-1}$ is continuous.

It would be nice, if I could prove the properties 2 and 4 also for $F^{-1+}$. Do they hold also for $F^{-1+}$? I needed property 3 to prove 4.

Best Answer

The function $F^{-1+}$ is continuous from the right. To see this let $y_{0}$ be such that $x_{0}:=F^{-1+}(y_{0})\in\mathbb{R}$ and consider a sequence $y_{n}\searrow y_{0}$. Set $F^{-1+}(y_{n}):=x_{n}$. Since $F^{-1+}$ is non-decreasing, $F^{-1+}(y_{0})\leq F^{-1+}(y_{n+1})\leq F^{-1+}(y_{n})$ and so $x_{0}\leq x_{n+1}\leq x_{n}$. It follows that $x_{n}\searrow x$ with $x_{0}\leq x$.

Since $x_{n}=F^{-1+}(y_{n})=\sup\{x:\,F(x)\leq y_{n}\}$ for every $\varepsilon>0$ we have $F(x_{n}-\varepsilon)\leq y_{n}<F(x_{n}+\varepsilon)$. If by contradiction $x_{0}<x$, then taking $\varepsilon:=\frac{x-x_{0}}{2}$ we have that $x_{n}-\varepsilon>x_{0}+\varepsilon$ and so by monotonicity $y_{n}\geq F(x_{n}-\varepsilon)\geq F(x_{0}+\varepsilon)>y_{0}$. Letting $n\rightarrow\infty$ and using the fact that $y_{n}\searrow y_{0}$ we get a contradiction. This shows that $x_{n}\searrow x_{0}$, that is, that $F^{-1+}$ is continuous from the right.

That $F^{-1+}$ admits a limit from the left follows from the fact that $F^{-1+}$ is non-decreasing. So (2) holds with right continuous in place of left.

Related Question