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Minimax adaptive tests for the Functional Linear model
"... Abstract:. 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 proc ..."
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Abstract:. 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 datadriven 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.
Adaptive Global Testing for Functional Linear Models
, 2013
"... This paper studies global testing of the slope function in functional linear regression models. A major challenge in functional global testing is to choose the dimension of projection when approximating the functional regression model by a finite dimensional multivariate linear regression model. We ..."
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This paper studies global testing of the slope function in functional linear regression models. A major challenge in functional global testing is to choose the dimension of projection when approximating the functional regression model by a finite dimensional multivariate linear regression model. We develop a new method that simultaneously tests the slope vectors in a sequence of functional principal components regression models. The sequence of models being tested is determined by the sample size and is an integral part of the testing procedure. Our theoretical analysis shows that the proposed method is uniformly powerful over a class of smooth alternatives when the signal to noise ratio exceeds the detection boundary. The methods and results reflect the deep connection between the functional linear regression model and the Gaussian sequence model. We also present an extensive simulation study and a real data example to illustrate the finite sample performance of our method.
ASYMPTOTIC EQUIVALENCE FOR INHOMOGENEOUS JUMP DIFFUSION PROCESSES AND WHITE NOISE.
"... Abstract. We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jumpdiffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and ..."
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Abstract. We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jumpdiffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and we suppose that the observation time T tends to ∞. The approximation is given in the sense of the Le Cam ∆distance, under smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other. 1.