Upper and lower quantiles

Question

I have a question regarding upper and lower quantiles.

In my lecture notes these are defined as:

$Q_p[X] = \inf\{x \mid P(X \leq x) \geq p\}$ is the lower p-quantile

$Q^p[X] = \inf\{x \mid P(X \leq x) > p\}$ is the upper p-quantile

For which distributions does $Q_p[X] = Q^p[X]$?

I can see that for continuous distributions with strictly increasing cdf the equality holds. But is it true for general continous r.v. $X$? If not, are there well known continous distributions where $Q_p[X] \neq Q^p[X]$

Answer 1

Here is a continuous but not strictly increasing CDF on the unit interval, defined in the obvious way:

We have

$$ Q_\frac{1}{2}[X] = \frac{1}{3} \quad\text{and}\quad Q^\frac{1}{2}[X] = \frac{2}{3}. $$

Whether this toy example is "well known" enough is of course debatable, but it can easily arise as the empirical CDF of a sample from a (hypothesized) continuous distribution.