Why use infimum in definition of Quantile function?

Question

This question was already asked here (I don't know if I should just bump that question instead of creating my own or how should I do it. If I did something wrong, I'm sorry.).

The Quantile function is defined as follows:

$$Q(p)=\inf\{x\in \mathbb{R}:p\leq F(x)\}, \hspace{.1cm} 0<p<1$$

But I don't really see an example of a distribution function for which $\inf$ can't be replaced with $\min$, because CDF is necessarily a right continuous function.

And I don't think this answer is correct, because CDF is always defined on $\mathbb{R}$, it shouldn't matter what values the random variable obtains.

Ian · Accepted Answer · 2021-12-08T18:10:03.540

7

Because $F$ is right-continuous and nondecreasing, the superlevel sets of $F$ are of the form $[a,\infty)$ where $a>-\infty$ or else the entire line. When the superlevel set is the whole line, there is no min (among the reals), while the inf is $-\infty$. For $a=+\infty$ the superlevel set is empty and so the inf is $+\infty$. These cases can potentially arise when $p=0$ or $p=1$ respectively; for $p \in (0,1)$ one can indeed replace the $\inf$ with $\min$.

edited Dec 08 '21 at 18:10

answered Sep 10 '16 at 14:22

Ian

101,645

If we define $A(r) :={x\lvert Pr(X\leq x) \geq r}$ does that then mean that for $r \in (0,1)$ open interval the inf could be changed to min? – Jesper Hybel Dec 08 '21 at 14:48
1

@JesperHybel Sure. I added a sentence clarifying this matter. – Ian Dec 08 '21 at 15:04
thx for the quick reply. That is helpful. – Jesper Hybel Dec 08 '21 at 15:13

Why use infimum in definition of Quantile function?

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