On super level-sets of conditional multivariate densities for multiple-output quantile regression
Kathrin Gruber (Erasmus School of Economics)
The simultaneous study of quantiles of multiple response variables requires a vector-valued approach to regression quantiles. However, common proposals of multivariate quantiles do not sufficiently control the probability content and thus, lack a clear probabilistic interpretation. We suggest superlevel-sets of conditional multivariate densities as an alternative multivariate quantile definition. Hence, the quantile is a function of the conditioning variables like in quantile regression. We show that such a conditional superlevel-set quantile has favorable mathematical and intuitive features.
We derive the quantile for a specified conditional or marginal density from an (overfitted) multivariate Gaussian mixture model to guarantee logically consistent (i.e., non-crossing) conditional quantile level sets. We demonstrate recovery of the true conditional quantiles for distributions with correlation, heteroskedasticity, or asymmetry and apply our method to a study on household expenditures.
Kathrin Gruber is assistant professor of econometrics at the Erasmus School of Economics.
Co-Authors: Dennis Fok, Annika Camehl
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