Laplace deconvolution

Abstract

We consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval. We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over a Laguerre functions basis using a regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. Because of this triangular structure, there is a common number m of terms in the function expansions to control, which is realized via a complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at T and provides an estimator with the risk within a logarithmic factor of m of the oracle risk. We emphasize that we consider the true observational model with possibly non-equispaced observations which are available on a finite interval of length T which appears in many different contexts, and we account for the bias associated with this model (which is not present in the case T -> infinity). The methodology is illustrated by an extensive simulation study and a real data example which confirms that the technique proposed is fast, efficient, accurate, usable from a practical point of view and very competitive. Joint with C.A. Cuenod, M. Pensky and Y. Rozenholc.