On superlevel sets of conditional densities and multivariate quantile regression

Some common proposals of multivariate quantiles do not sufficiently control the probability content, while others do not always accurately reflect the concentration of probability mass. We suggest superlevel sets of conditional multivariate densities as an alternative to current multivariate quantile definitions. Hence, the superlevel set is a function of conditioning variables much like in quantile regression. We show that conditional superlevel sets have favorable mathematical and intuitive features, and support a clear probabilistic interpretation. We derive the superlevel sets for a conditional or marginal density of interest from an (overfitted) multivariate Gaussian mixture model. This approach guarantees logically consistent (i.e., non-crossing) conditional superlevel sets and also allows us to obtain more traditional univariate quantiles. We demonstrate recovery of the true conditional univariate quantiles for distributions with correlation, heteroskedasticity, or asymmetry and apply our method in univariate and multivariate settings to a study on household expenditures.