The Annals of Statistics

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

Guenther Walther

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 20 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1069362379

Digital Object Identifier
doi:10.1214/aos/1069362379

Mathematical Reviews number (MathSciNet)
MR1474075

Zentralblatt MATH identifier
0881.62067

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

Keywords
Halfspace distance dual measure random search central limit theorem Monte Carlo

Citation

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. https://projecteuclid.org/euclid.aos/1069362379


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