Laplace deconvolution with dependent errors: a minimax study

Abstract

ABSTRACTWe investigate the problem of estimating a function f based on observations from its noisy convolution when the noise exhibits long-range dependence (LRD). We consider both Gaussian and sub-Gaussian errors. We construct an adaptive estimator based on the kernel method, with the optimal selection of the bandwidths performed via Lepski's Method. We derive a minimax lower bound for the-risk when f belongs to a Sobolev ball and show that such estimator attains optimal or near-optimal rates that deteriorate as the LRD worsens. We carry out a limited simulations study which confirms our conclusions from theoretical results.