The Annals of Statistics

On multivariate quantiles under partial orders

Alexandre Belloni and Robert L. Winkler

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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.

Article information

Ann. Statist., Volume 39, Number 2 (2011), 1125-1179.

First available in Project Euclid: 9 May 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62J99: None of the above, but in this section
Secondary: 62J07: Ridge regression; shrinkage estimators

Multivariate quantiles partial order uniform estimation


Belloni, Alexandre; Winkler, Robert L. On multivariate quantiles under partial orders. Ann. Statist. 39 (2011), no. 2, 1125--1179. doi:10.1214/10-AOS863.

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