The Annals of Statistics

Granulometric smoothing

Guenther Walther

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Abstract

A new method for smoothing a multivariate data set is introduced that is based on a simple geometric operation. This method is applied to the problem of estimating level sets of a density and minimum volume sets with given probability content. Building on existing techniques, the resulting estimator combines excellent theoretical and computational properties: It converges with the minimax rates (up to log factors) in most cases where these rates are known and, at the same time, it can be computed, visualized, stored and manipulated by simple algorithms and tools. It is applicable to a wide class of sets that is characterized explicitly in terms of the underlying densities and includes nonconvex and disconnected sets, and it is argued that it should give reasonable results in completely general situations. Applications to the construction of multivariate confidence regions in frequentist and Bayesian contexts are briefly mentioned.

Article information

Source
Ann. Statist., Volume 25, Number 6 (1997), 2273-2299.

Dates
First available in Project Euclid: 30 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1030741072

Digital Object Identifier
doi:10.1214/aos/1030741072

Mathematical Reviews number (MathSciNet)
MR1604445

Zentralblatt MATH identifier
0919.62026

Subjects
Primary: 62G05: Estimation
Secondary: 65U05

Keywords
Granulometry smoothing level sets minimum volume sets excess mass mode multivariate confidence regions highest posterior density regions

Citation

Walther, Guenther. Granulometric smoothing. Ann. Statist. 25 (1997), no. 6, 2273--2299. doi:10.1214/aos/1030741072. https://projecteuclid.org/euclid.aos/1030741072


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