On multivariate quantiles under partial orders

Abstract

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of {\it partial quantiles} based on a given partial order. We establish that partial quantiles are equivariant under partial-order-preserving transformations of the data, display a concentration of measure phenomenon, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, and establish computational complexity bounds. Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets and to evaluate the performance of investment funds are presented.