Statistica Sinica 18(2008), 255-279

THE MAXISET POINT OF VIEW FOR ESTIMATING

INTEGRATED QUADRATIC FUNCTIONALS

Vincent Rivoirard and Karine Tribouley

CNRS UMR 8628, Paris XI-Orsay and Modal'X, Paris X-Nanterre

Abstract: This paper explores the estimation of $\int\! f^2$ where $f$ is a functional parameter in the white noise model. To compare different estimation procedures, we adopt the maxiset point of view, i.e., we point out the entire set of functions on which a given procedure achieves a given target rate. Quadratic and soft (local and global) thresholding wavelet procedures are considered. We compute the maxisets for these procedures and prove that, most of the time, thresholding procedures outperform the quadratic one. The comparison of performances in the maxiset setting of local and global thresholding depends on the target rate; none of them is always preferable.

Key words and phrases: Besov spaces, integrated functional, maxiset theory, minimax theory, thresholding, wavelet.