Xi'an's Og

My 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.

While 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.

This entry was posted on July 4, 2014 at 12:14 am and is filed under pictures, Statistics, University life with tags CEREMADE, Monte Carlo methods, multivariate quantile, nonparametrics, Paris, quantile regression, simulation, Université Paris Dauphine. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.