Electronic Journal of Statistics

A powerful test based on tapering for use in functional data analysis

Dan J. Spitzner

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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 QnOPT=nj=1pnj1/2Yn,j2. Down-weighting by j1/2 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 QnOPT 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.

Article information

Electron. J. Statist., Volume 2 (2008), 939-962.

First available in Project Euclid: 8 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62J05: Linear regression
Secondary: 46N30: Applications in probability theory and statistics

functional data analysis quadratic forms high-dimensional testing rates of testing Fourier decomposition


Spitzner, Dan J. A powerful test based on tapering for use in functional data analysis. Electron. J. Statist. 2 (2008), 939--962. doi:10.1214/08-EJS172. https://projecteuclid.org/euclid.ejs/1223477211

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