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2008 A powerful test based on tapering for use in functional data analysis
Dan J. Spitzner
Electron. J. Statist. 2: 939-962 (2008). DOI: 10.1214/08-EJS172

Abstract

A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphasize the smooth attributes of the model. A formula is $Q_n^{OPT}=n∑_{j=1}^{p_n}j^{−1/2}‖𝒀 _{n,j}‖^2$. Down-weighting by $j^{−1/2}$ is selected to achieve adaptive optimality among tests based on tapering with respect to its “rates of testing,” an asymptotic framework for measuring a test’s retention of power in high dimensions under smoothness constraints. Existing tests based on truncation or thresholding are known to have superior asymptotic power in comparison with any test based on tapering; however, it is shown here that high-order effects can be substantial, and that a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the sort of alternatives that would typically be of concern in functional data analysis applications. The proposed test is developed for use in practice, and demonstrated in an example application.

Citation

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Dan J. Spitzner. "A powerful test based on tapering for use in functional data analysis." Electron. J. Statist. 2 939 - 962, 2008. https://doi.org/10.1214/08-EJS172

Information

Published: 2008
First available in Project Euclid: 8 October 2008

zbMATH: 1320.62104
MathSciNet: MR2447346
Digital Object Identifier: 10.1214/08-EJS172

Subjects:
Primary: 62G10 , 62J05
Secondary: 46N30

Keywords: Fourier decomposition , Functional data analysis , high-dimensional testing , Quadratic forms , rates of testing

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society

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