Data Depth and Multiple Output Regression, the Distorted M-Quantiles Approach

For a univariate distribution, its M-quantiles are obtained as solutions to asymmetric minimization problems dealing with the distance of a random variable to a fixed point. The asymmetry refers to the different weights awarded to the values of the random variable at either side of the fixed point. We focus on M-quantiles whose associated losses are given in terms of a power. In this setting, the classical quantiles are obtained for the first power, while the expectiles correspond to quadratic losses. The M-quantiles considered here are computed over distorted distributions, which allows to tune the weight awarded to the more central or peripheral parts of the distribution. These distorted M-quantiles are used in the multivariate setting to introduce novel families of central regions and their associated depth functions, which are further extended to the multiple output regression setting in the form of conditional and regression regions and conditional depths.This research was partially funded by the Spanish Ministry of Science and Innovation under grant PID2021-123592OB-I00 and the V Regional Plan for Scientific Research and Technological Innovation 2016–2020 of the Community of Madrid, an agreement with Universidad Carlos III de Madrid in the action of “Excellence for University Professors”