Noisy Laplace deconvolution with error in the operator

• We consider Laplace deconvolution with random noise in a regression framework.
• The convolution kernel is unknown, and accessible only through experimental noise.
• We build two thresholding procedures, adaptive to the target function but not to the blurring kernel.
• We prove their optimality in a minimax sense.

We address the problem of Laplace deconvolution on R+R+ in a white noise framework. The convolution kernel is unknown, and accessible only through experimental noise. We make use of a recent procedure of estimation based on a Galerkin projection of the operator on Laguerre functions (Comte et al., 2012), and couple it with a thresholding procedure performed both on the noisy kernel and on the noisy convoluted signal. We establish the minimax optimality of our procedure under the squared loss error, when the smoothness of the signal is measured in a Laguerre–Sobolev sense and the kernel satisfies standard blurring assumptions. The resulting process is adaptive with respect to the target function’s smoothness, but not to the unknown degree of ill-posedness of the operator. We conclude this paper with a numerical study emphasizing the good practical performances of the procedure on concrete examples.