## vector quantile regression

**M**y Paris-Dauphine colleague Guillaume Carlier recently arXived a statistics paper entitled *Vector quantile regression*, co-written with Chernozhukov and Galichon. I was most curious to read the paper as Guillaume is primarily a mathematical analyst working on optimisation problems like optimal transport. And also because I find quantile regression difficult to fathom as a statistical problem. (As it happens, both his co-authors are from econometrics.) The results in the paper are (i) to show that a d-dimensional (Lebesgue) absolutely continuous random variable Y can always be represented as the deterministic transform Y=Q(U), where U is a d-dimensional [0,1] uniform (the paper expresses this transform as conditional on a set of regressors Z, but those essentially play no role) and Q is monotonous in the sense of being the gradient of a convex function,

and

(ii) to deduce from this representation a unique notion of multivariate quantile function; and (iii) to consider the special case when the quantile function Q can be written as the linear

where β(U) is a matrix. Hence leading to an estimation problem.

**W**hile unsurprising from a measure theoretic viewpoint, the representation theorem (i) is most interesting both for statistical and simulation reasons. Provided the function Q can be easily estimated and derived, respectively. The paper however does not provide a constructive tool for this derivation, besides indicating several characterisations as solutions of optimisation problems. From a statistical perspective, a non-parametric estimation of β(.) would have useful implications in multivariate regression, although the paper only considers the specific linear case above. Which solution is obtained by a discretisation of all variables and linear programming.

July 8, 2014 at 6:10 pm

“And also because I find quantile regression difficult to fathom as a statistical problem.”

hi christian, i was wondering what you meant by this comment. could you elaborate? also, more generally, what do you think of as a “statistical problem”? thanks in advance.

July 9, 2014 at 8:11 am

(i) I do not understand the practical purpose of estimating a single quantile function rather compared with learning the entire conditional cdf. In the sense that I cannot see a case where a single quantile would be of interest. Except maybe in an extremely formalised regulation setting like Phase xxx medical trial…

(ii) As to the very nature of statistical problems, this is a fairly subjective perspective and feeling!