The Annals of Statistics

An algorithmic and a geometric characterization of coarsening at random

Richard D. Gill and Peter D. Grünwald

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We show that the class of conditional distributions satisfying the coarsening at random (CAR) property for discrete data has a simple and robust algorithmic description based on randomized uniform multicovers: combinatorial objects generalizing the notion of partition of a set. However, the complexity of a given CAR mechanism can be large: the maximal “height” of the needed multicovers can be exponential in the number of points in the sample space. The results stem from a geometric interpretation of the set of CAR distributions as a convex polytope and a characterization of its extreme points. The hierarchy of CAR models defined in this way could be useful in parsimonious statistical modeling of CAR mechanisms, though the results also raise doubts in applied work as to the meaningfulness of the CAR assumption in its full generality.

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2409-2422.

First available in Project Euclid: 13 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics
Secondary: 62N01: Censored data models

Coarsening at random (CAR) ignorability uniform multicover Fibonacci numbers


Gill, Richard D.; Grünwald, Peter D. An algorithmic and a geometric characterization of coarsening at random. Ann. Statist. 36 (2008), no. 5, 2409--2422. doi:10.1214/07-AOS532.

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