Laplace deconvolution with noisy observations

Abstract

In the present paper we consider Laplace deconvolution problem for discrete noisy data observed on an interval whose length $T_{n}$ may increase with the sample size. Although this problem arises in a variety of applications, to the best of our knowledge, it has been given very little attention by the statistical community. Our objective is to fill the gap and provide statistical analysis of Laplace deconvolution problem with noisy discrete data. The main contribution of the paper is an explicit construction of an asymptotically rate-optimal (in the minimax sense) Laplace deconvolution estimator which is adaptive to the regularity of the unknown function. We show that the original Laplace deconvolution problem can be reduced to nonparametric estimation of a regression function and its derivatives on the interval of growing length $T_{n}$. Whereas the forms of the estimators remain standard, the choices of the parameters and the minimax convergence rates, which are expressed in terms of $T_{n}^{2}/n$ in this case, are affected by the asymptotic growth of the length of the interval.

We derive an adaptive kernel estimator of the function of interest, and establish its asymptotic minimaxity over a range of Sobolev classes. We illustrate the theory by examples of construction of explicit expressions of Laplace deconvolution estimators. A simulation study shows that, in addition to providing asymptotic optimality as the number of observations tends to infinity, the proposed estimator demonstrates good performance in finite sample examples.