Bootstrap in functional linear regression

“…As for estimation, setting k is arguably a difficult problem. To bypass this calibration issue, one may apply a permutation approach [8] or use bootstrap methodologies [15,21]. While the levels of the corresponding tests are asymptotically controlled, there is again no theoretical guarantee on the power.…”

Section: Introductionmentioning

confidence: 99%

Minimax adaptive tests for the functional linear model

Hilgert¹,

Mas²,

Verzélen³

2013

Ann. Statist.

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. 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.

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“…give generalizations to strong mixing processes. González-Manteiga and Martínez-Calvo (2011) study the naive and the wild bootstrap in a linear regression model for FPCA-type estimates.…”

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confidence: 99%

The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions

Krebs

2019

Journal of Multivariate Analysis

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We consider the double functional nonparametric regression model Y = r(X)+ε, where the response variable Y is Hilbert space-valued and the covariate X takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields indexed by the regular lattice Z N for some N ∈ N+.We consider a kernel estimator of the Nadaraya-Watson type for the regression operator r and study its limiting law which is a Gaussian operator on the Hilbert space. Moreover, we investigate both a naive and a wild bootstrap procedure in the double functional setting and demonstrate their asymptotic validity. This is quite useful as building confidence sets based on an asymptotic Gaussian distribution is often difficult.

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Bootstrap in functional linear regression

Cited by 31 publications

References 15 publications

A survey of functional principal component analysis

A survey of functional principal component analysis

Minimax adaptive tests for the functional linear model

The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions

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