I have seen the definition of quantile function here, which is as follows (slightly modified):

Let $X$ be a real-valued non-degenerate random variable with distribution function $F_X(x)=\mathbb{P}({X\leq x}).$ The inverse or quantile function $Q_X$ of $F_X$ is defined as:

$$Q_X(y) = \inf \left\{x\in\mathbb{R}:F_{X}(x)\geq y\right\} $$

where $0 < y \leq 1$ , and$\quad Q_X ( 0 ) = \inf \{ x \in \mathbb { R } : F_X( x ) > 0 \}$

My question is:

  1. Why don't we define the lower endpoint of the range of $y$ as $Q_{X}(0) = -\infty$

  2. Why don't we also define the upper endpoint of the range of $y$ of $Q_{X}(1) = \infty$

EDITED: deleted "if $F_{X}(x)<1$" in "2. Why don't we also define the upper endpoint of the range of y of $Q_{X}(1) = \infty \;\text{if} \; F_{X}(x)<1$"

  • 1
    $\begingroup$ Does the explanation at en.wikipedia.org/wiki/Quantile_function#Definition help? $\endgroup$
    – A. Donda
    Commented Nov 24, 2018 at 15:53
  • $\begingroup$ @A. Donda, wikipedia's explanation helps, but it leaves a few subtle points out: the domain of $F_X$ is $[0,1]$, but the domain of $Q_X$ is $(0,1)$ as $0<p<1$. Thus, it does not explicitly say what happens at the end points, $p=0$ and $p=1$. $\endgroup$
    – AlexMe
    Commented Nov 24, 2018 at 16:16

1 Answer 1

  1. Fixing $Q_X(0)$ at $-\infty$ would make it completely uninformative when it doesn't need to be. Consider, for example, a finite support distribution such as the uniform between $a$ and $b$. By the definition of $Q_X$ you saw, that would make $Q_X(0)=a$, which tells us that the possible values of $X$ begin at $a$ (which is at least some information). enter image description here $Q_X(0)=-\infty$ would then tell us that the density (or probability if it is not continuous) function of $X$ has an infinite tail going left.
    1. For the same reason really. Tell me if you want me to expand on this.
  • $\begingroup$ many thanks @Bananin, your explanation definitely made this notation more clear! I guess the notation I saw is intended to cover two possible type of domains - bounded and unbounded. Yes, please, it would be helpful if you could expand 2. and if it is possible to comment why would the domain be $(0,1]$ and not $(0,1)$, in which case we could equally define $Q_X(1)=\text{sup}\{x \in \mathbb{R}: F_X(x)<1\}$ $\endgroup$
    – AlexMe
    Commented Nov 25, 2018 at 1:22
  • 1
    $\begingroup$ For 2. , in the example I gave $Q_X(1)=b$, so $Q_X(1)$ tells us where the possible values of $X$ end on the real line. I think your observation that the definition is meant to cover bounded as well as unbounded domains is the key to questions 1 and 2. With regards to the question about the $sup$, I think your definition would be the same as the one with $inf$ because the sets ${x:F_X(x)<1}$ and ${x:F_X(x)\geq 1}$ are right next to each other. In terms of the domain, $Q_X$ is defined on $(0,1)$. It being defined on $0$ or $1$ depends on the density function being bounded on the left or right. $\endgroup$
    – Bananin
    Commented Nov 25, 2018 at 22:28

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