First some standard notation. A probability triplet is $(\Omega,\mathcal F, P)$. You have a random variable $X : \Omega \to \mathbb{R}$ measurable function. The distribution function is $F(x) = P(X\leq x) = P(\{\omega\in \Omega : X(\omega) \leq x\})$ We know $F$ has left-limits and is right-continous.
Below I give three notions of quantiles, and then finally I state the question.
Standard quantile function
Wikipedia states that the quantile function $F^{-1}:(0,1)\to\mathbb R$ defined by $$ F^{-1}(a) = \inf \{ x \in \mathbb{R} : F(x) \geq a \}$$ is the unique function that fulfils $$ a \leq F(x) \quad\text{if and only if}\quad F^{-1}(a) \leq x$$ and I can see the naturalness in that condition. Since we are not guaranteed an inverse of $F$ we settle with the next best thing: an almost-inverse that preserves partial order.
This is the only definition of a quantile function I've seen, and it seems natural.
"A quantile"
The second variant I have seen is from Allan Gut (2013) Probability, a Graduate Course. Similar definitions can be found in other works. Definition 7.1, p140 says (after replacing some symbols)
The $b$-quantile $\lambda_b(X)$ of a random variable $X$ is a real number that satisfies $$P(X\geq \lambda_b(X))\geq b$$
Remark: The median is thus a 1/2-quantile
I understand the definition as meaning "all" real numbers satisfying the inequality. This is useful for colloquialisms as "this falls in the 1/2-quantile" if a value is below the median, as the definition from above. The definition is awkward since it looks at right-tail probabilites, rather than left-tail ones, but that is a minor thing. In Exercise 11.2, p249, where $X$ is a standard normal, he makes this explicit by mentioning in passing that $F(\lambda_b)=1-b$.
"Quantile sets"
In Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. "Coherent Measures of Risk". Mathematical Finance 9, 203–228 (1999), yet another variant is proposed.
In this article, they first stipulate that $\Omega$ is finite. I guess this is to make sure that limits always exists and so on.... But anyway. Definition 3.2, after replacing some symbols, states that
Given $c \in (0,1)$, the number $q$ is a $c$-quantile of the random variable $X$ under distribution $P$ is any of the following three conditions are satisfied
i. $P(X\leq q)\geq c$ and $c \geq P(X< q)$
ii. $P(X\leq q)\geq c$ and $ P(X\geq q) \geq 1-c$
iii. $F(q)\geq c$ and $c \geq F(q_-)$ where $F(q_-) = \lim_{x\to q,r<q}F(x)$
Remark: The set of such $c$-quantiles is a closed interval. Since $\Omega$ is finite, there is a finite left (respectively, right) end point $q^{-}_{c}$ (respectively, $q^{+}_{c}$) that satisfies $q^{-}_{c} = \inf \{ x : F(x) \geq a \}$ [equivalenty $\sup \{x:F(x)<c\}$] (resp. $q^{+}_{c} = \inf \{ x : F(x) > c \}$. With the exception of at most countably many $c$, the equality $q^{-}_{c}=q^{+}_{c}$ holds. [...]
My question
These three notions share a lot, but they handle edge cases quite differently. For continous, strictly monotonous $F$, the differences are not too interesting, but they work out differently where $F$ has jumps and plateaus. And it bugs me, since I am working with a discrete distribution, and then $F$ is jumps and plateaues everywhere.
Is there a unifying view that gathers all these three variants of quantiles, or that motivates their differences?
Is there a way to talk about left- and right-quantiles within the framework of the standard quantile function?
Is there any source you consider authorative on the definition of quantiles in various forms?
Close duplicates
Is there one universally agreed-upon definition of a quantile? If not what definitions exist? is a similar question is raised, but solely from a practical/software perspective. I am rather looking for a mathematical discussion.
Are there two different possible meanings for a quantile group? is confused about various meanings of quantile groups, but I am interested in point quantiles.
Definition of quantile is only about sample quantiles.
