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October 1997 Monte Carlo sampling in dual space for approximating the empirical halfspace distance
Guenther Walther
Ann. Statist. 25(5): 1926-1953 (October 1997). DOI: 10.1214/aos/1069362379


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.


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Guenther Walther. "Monte Carlo sampling in dual space for approximating the empirical halfspace distance." Ann. Statist. 25 (5) 1926 - 1953, October 1997.


Published: October 1997
First available in Project Euclid: 20 November 2003

zbMATH: 0881.62067
MathSciNet: MR1474075
Digital Object Identifier: 10.1214/aos/1069362379

Primary: 62G05
Secondary: 62H40 , 65U05

Keywords: central limit theorem , dual measure , halfspace distance , Monte Carlo , random search

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 5 • October 1997
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