Open Access
April 2020 Multidimensional multiscale scanning in exponential families: Limit theory and statistical consequences
Claudia König, Axel Munk, Frank Werner
Ann. Statist. 48(2): 655-678 (April 2020). DOI: 10.1214/18-AOS1806


We consider the problem of finding anomalies in a $d$-dimensional field of independent random variables $\{Y_{i}\}_{i\in \{1,\ldots,n\}^{d}}$, each distributed according to a one-dimensional natural exponential family $\mathcal{F}=\{F_{\theta }\}_{\theta \in \Theta }$. Given some baseline parameter $\theta _{0}\in \Theta $, the field is scanned using local likelihood ratio tests to detect from a (large) given system of regions $\mathcal{R}$ those regions $R\subset \{1,\ldots,n\}^{d}$ with $\theta _{i}\neq \theta _{0}$ for some $i\in R$. We provide a unified methodology which controls the overall familywise error (FWER) to make a wrong detection at a given error rate.

Fundamental to our method is a Gaussian approximation of the distribution of the underlying multiscale test statistic with explicit rate of convergence. From this, we obtain a weak limit theorem which can be seen as a generalized weak invariance principle to nonidentically distributed data and is of independent interest. Furthermore, we give an asymptotic expansion of the procedures power, which yields minimax optimality in case of Gaussian observations.


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Claudia König. Axel Munk. Frank Werner. "Multidimensional multiscale scanning in exponential families: Limit theory and statistical consequences." Ann. Statist. 48 (2) 655 - 678, April 2020.


Received: 1 March 2018; Revised: 1 August 2018; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241564
MathSciNet: MR4102671
Digital Object Identifier: 10.1214/18-AOS1806

Primary: 60F17 , 62H10
Secondary: 60G50 , 62F03

Keywords: exponential families , familywise error rate , invariance principle , multiscale testing , scan statistic , weak limit

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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