1 Exordium

These notes are intended as a place where I’ll gather interesting facts about quantiles and quantile regression (from elementary to fringe). This is definitely not a textbook: for that you can consult Koenker (2005) and Koenker et al. (2017). In any case I have included many exercises that clarify curious and relevant facts that appear scattered in the literature. This notes are a work in progress, and likely will always be: whenever I deem a topic worthy of appearing here, I’ll add it sooner or later. The github repository for the notes can be found at https://github.com/eduardohorta/QuantRegFacts.

It is assumed that the reader has some knowledge of Probability and Statistics. The expression “measurable function” appears many times in the text; if you don’t know what it means, substitute “measurable” by “piecewise continuous.” You won’t lose much.

In the Probability and Statistics literature, it has become a habit to adopt the notational abuse of writing \(g(X)\) when one really means the composition \(g \circ X.\) There is no easy way around,1 and sometimes ambiguity does arise.2 At any rate, in most cases one can tell from context what is the correct interpretation, that is, in writing \(g(X)\) we usually know beforehand what the symbol \(g\) stands for. For example, if \(X\) is a random variable and \(g\) is a real valued measurable function on \(\mathbb{R},\) then \(g(X)\) stands for \(g\circ X.\) Similarly, if \(g\) is a functional or operator, say \(g = \mathbf{E},\) then \(\mathbf{E}(X)\) is not a composition but evaluation of the functional \(\mathbf{E}\) at the “point” \(X.\) In view of this, I shall (in dismay) go along with tradition.

  1. We could give up tradition and go on with the more correct \(g\circ X\) but the notation can get clumsy, especially if \(g\) is a function of many variables. Moreover, tradition exists — at least in part — for good reasons we do not fully understand.↩︎

  2. For instance, for a random vector \(X\) the notation \(\Vert X\Vert_2\) can be used both for the “random Euclidean norm” \(\Vert X\Vert_2 = \sqrt{\sum\nolimits_{d=1}^{\mathrm{D}_X}X_d^2}\) and for the proper \(L^2\) norm, \(\Vert X\Vert_2 = \mathbf{E}\sqrt{\sum\nolimits_{d=1}^{\mathrm{D}_X}X_d^2}.\)↩︎