Title: Anisotropic functional Laplace deconvolution,
Abstract: In the present paper we consider the problem of estimating a
three-dimensional function $f$ based on observations from its noisy Laplace
convolution. Our study is motivated by the analysis of Dynamic Contrast
Enhanced (DCE) imaging data. We construct an adaptive wavelet-Laguerre
estimator of $f$, derive minimax lower bounds for the $L^2$-risk when $f$
belongs to a three-dimensional Laguerre-Sobolev ball and demonstrate that the
wavelet-Laguerre estimator is adaptive and asymptotically near-optimal in a
wide range of Laguerre-Sobolev spaces. We carry out a limited simulations study
and show that the estimator performs well in a finite sample setting. Finally,
we use the technique for the solution of the Laplace deconvolution problem on
the basis of DCE Computerized Tomography data.