Directional M-quantile regression for multivariate dependent outcomes

In the present work we generalize the univariate M-quantile regression to the analysis of multivariate dependent outcomes. Extending the notion of directional quantiles, we introduce directional M-quantiles which are obtained as projections of the original data on a specified unit norm direction. In order to take into consideration the correlation within grouped measurements and to increase efficiency, we develop a marginal M-Quantile regression model extending the well known generalized estimating equations approach. We build M-quantile regions and contours which allow us to investigate the effect of the covariates on the location, spread and shape of the distribution of the responses. To identify potential outliers and provide a simple visual representation of the variability of the M quantile contours estimator, we construct confidence envelope via nonparametric bootstrap. The validity of our method is analyzed through the study of the wages data from the National Longitudinal Survey of Youth.