Minimax adaptive tests for the Functional Linear model
By Nadine Hilgert, André Mas, Nicolas Verzelen. 2012
68 citations recorded by the source corpus.
We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the procedures are completely data-driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable \log\log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.
Subfields: Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/4f896c7c7b0e298fa306962cc683d5b6381aea65
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In Statistics and Probability
Citations in the corpus
- Functional Data Analysis (2016, background): “…the coefficient vector β by a centered functional covariate X c = X ( t ) − µ ( t ) and coefficient function β = β ( t ), for t ∈ I , one arrives at the functional linear model which has been studied extensively (Cardot, Ferraty & Sarda 1999, 2003; Hall & Horowitz 2007; Hilgert, Mas & Verzelen 2013).”
- Adaptive Global Testing for Functional Linear Models (2014, background): “Hilgert, Mas, and Verzelen (2012) independently considered the global testing under the functional linear model with a different alternative hypothesis ‖ 1/2θ‖2 ≥ r ′n (instead of ‖θ‖2 ≥ rn considered in this article).”
- Nonparametric inference in generalized functional linear models (2014, methodology): “We conclude this subsection by showing that the proposed adaptive test can still achieve the optimal minimax rate (up to a logarithmic order) specified in [15], that is, δ(n,k), even under non-Gaussian errors.”