The Annals of Statistics

Monte Carlo sampling in dual space for approximating the empirical halfspace distance

Guenther Walther

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The Kolmogorov-Smirnov distance is an important tool for constructing confidence sets and tests in univariate problems. In multivariate settings, an analogous role is played by the halfspace distance, which has the merit of being invariant under linear transformations. However, the evaluation of the halfspace distance between two samples is a computationally very intensive combinatorial problem even in moderate dimensions, which severely restricts the use of the halfspace distance, especially in resampling procedures. To approximate this distance in a fast and data-dependent way, the notion of a dual measure is introduced. Based on geometric concepts, it will be shown how the above problem can be put as a density estimation problem using Monte Carlo sampling in a certain dual space. A central limit theorem for the empirical halfspace distance is derived and used as a gauge to compare the new procedure with a traditional random search.

Article information

Ann. Statist., Volume 25, Number 5 (1997), 1926-1953.

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62H40 65U05

Halfspace distance dual measure random search central limit theorem Monte Carlo


Walther, Guenther. Monte Carlo sampling in dual space for approximating the empirical halfspace distance. Ann. Statist. 25 (1997), no. 5, 1926--1953. doi:10.1214/aos/1069362379.

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