Multivariate quantile regression using superlevel sets of conditional densities

In multiple-output quantile regression the simultaneous study of multiple response variables requires multivariate quantiles. Current definitions of such quantiles often lack a clear probability interpretation, as the defined quantiles can cover large parts of the distribution where little probability mass is located or their enclosed area does not equal the quantile level. We suggest superlevel-sets of conditional multivariate density functions as an alternative multivariate quantile definition. Such a quantile set contains all points in the domain for which the density exceeds a certain level. By applying this to a conditional density, the quantile becomes a function of the conditioning variables. We show that such a quantile has favorable mathematical and intuitive features. For implementation, we, first, use an overfitted Gaussian mixture model to fit the multivariate density and, next, calculate the multivariate quantile for a conditional or marginal density of interest. Operating on the same estimated multivariate density guarantees logically consistent quantiles. In particular, the quantiles at multiple percentiles are non-crossing. We use simulation to demonstrate that we recover the true quantiles for distributions with correlation, heteroskedasticity, or asymmetry in the disturbances and we apply our method to study heterogeneity in household expenditures.