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The block criterion for multiscale inference about a density, with applications to other multiscale problems


Rufibach, K; Walther, G (2010). The block criterion for multiscale inference about a density, with applications to other multiscale problems. Journal of Computational and Graphical Statistics, 19(1):175-190.

Abstract

The use of multiscale statistics, i.e. the simultaneous inference about various stretches
of data via multiple localized statistics, is a natural and popular method for inference about
e.g. local qualitative characteristics of a regression function, a density, or its hazard rate. We
focus on the problem of providing simultaneous confidence statements for the existence of
local increases and decreases of a density and address several statistical and computational
issues concerning such multiscale statistics. We first review the benefits of employing scaledependent
critical values for multiscale statistics and then derive an approximation scheme
that results in a fast algorithm while preserving statistical optimality properties. The main
contribution is a methodology for calibrating multiscale statistics that does not require a caseby-
case derivation of its specific form. We show that in the above density context the methodology
possesses statistical optimality properties and allows for a fast algorithm. We illustrate
the methodology with two further examples: A multiscale statistic introduced by Gijbels and
Heckman for inference about a hazard rate and local rank tests introduced by D¨umbgen for
inference in nonparametric regression.
Code for the density application is available as the R package modehunt on CRAN. Additional
code to compute critical values, reproduce the hazard rate and local rank example and
the plots in the paper as well as datasets containing simulation results and an appendix with
all the proofs of the theorems are available online as supplemental material.

The use of multiscale statistics, i.e. the simultaneous inference about various stretches
of data via multiple localized statistics, is a natural and popular method for inference about
e.g. local qualitative characteristics of a regression function, a density, or its hazard rate. We
focus on the problem of providing simultaneous confidence statements for the existence of
local increases and decreases of a density and address several statistical and computational
issues concerning such multiscale statistics. We first review the benefits of employing scaledependent
critical values for multiscale statistics and then derive an approximation scheme
that results in a fast algorithm while preserving statistical optimality properties. The main
contribution is a methodology for calibrating multiscale statistics that does not require a caseby-
case derivation of its specific form. We show that in the above density context the methodology
possesses statistical optimality properties and allows for a fast algorithm. We illustrate
the methodology with two further examples: A multiscale statistic introduced by Gijbels and
Heckman for inference about a hazard rate and local rank tests introduced by D¨umbgen for
inference in nonparametric regression.
Code for the density application is available as the R package modehunt on CRAN. Additional
code to compute critical values, reproduce the hazard rate and local rank example and
the plots in the paper as well as datasets containing simulation results and an appendix with
all the proofs of the theorems are available online as supplemental material.

Citations

3 citations in Web of Science®
3 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:04 Faculty of Medicine > Epidemiology, Biostatistics and Prevention Institute (EBPI)
Dewey Decimal Classification:610 Medicine & health
Language:English
Date:2010
Deposited On:26 Mar 2010 11:41
Last Modified:05 Apr 2016 14:04
Publisher:American Statistical Association
ISSN:1061-8600
Publisher DOI:https://doi.org/10.1198/jcgs.2009.07071
Permanent URL: https://doi.org/10.5167/uzh-33106

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