On multivariate quantiles under partial orders

TL;DR

This paper introduces partial quantiles based on partial orders, providing a robust, order-preserving framework for multivariate data analysis, with theoretical properties, estimators, and practical applications.

Contribution

It generalizes univariate quantiles to multivariate settings using partial orders, establishing their properties, estimation methods, and computational aspects.

Findings

Partial quantiles are equivariant under order-preserving transformations.

They are robust to outliers and can characterize distributions.

Applications include diet analysis, investment performance, and policy impact studies.

Abstract

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, 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, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.