Open Access
Translator Disclaimer
April 2011 On multivariate quantiles under partial orders
Alexandre Belloni, Robert L. Winkler
Ann. Statist. 39(2): 1125-1179 (April 2011). DOI: 10.1214/10-AOS863


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.


Download Citation

Alexandre Belloni. Robert L. Winkler. "On multivariate quantiles under partial orders." Ann. Statist. 39 (2) 1125 - 1179, April 2011.


Published: April 2011
First available in Project Euclid: 9 May 2011

zbMATH: 1216.62082
MathSciNet: MR2816350
Digital Object Identifier: 10.1214/10-AOS863

Primary: 62H12 , 62J99
Secondary: 62J07

Keywords: Multivariate quantiles , partial order , uniform estimation

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 2 • April 2011
Back to Top