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Optimal testing in a fixed-effects functional analysis of variance model. (English) Zbl 1071.62037

Summary: We consider the testing problem in a fixed-effects functional analysis of variance model. We test the null hypotheses that the functional main effects and the functional interactions are zeros against the composite nonparametric alternative hypotheses that they are separated away from zero in \(L^2\)-norm and also possess some smoothness properties. We adapt the optimal (minimax) hypothesis testing procedures for testing a zero signal in a Gaussian “signal plus noise” model to derive optimal (minimax) non-adaptive and adaptive hypothesis testing procedures for the functional main effects and the functional interactions. The corresponding tests are based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize different types of smoothness conditions assumed on the response function by means of its wavelet coefficients for a wide range of function classes.
In order to shed some light on the theoretical results obtained, we carry out a simulation study to examine the finite sample performance of the proposed functional hypothesis testing procedures. As an illustration, we also apply these tests to a real-life data example arising from physiology. Concluding remarks and hints for possible extensions of the proposed methodology are also given.

MSC:

62G10 Nonparametric hypothesis testing
46N30 Applications of functional analysis in probability theory and statistics
62J10 Analysis of variance and covariance (ANOVA)
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

Besov spaces

Software:

fda (R); gss
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Full Text: DOI

References:

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