Adaptive local density estimation in tomography

TL;DR

This paper develops a data-driven spectral cut-off estimator for non-parametric density estimation in tomography, achieving minimax optimality and demonstrating good practical performance through simulations.

Contribution

It introduces a fully data-driven spectral cut-off regularisation method for density estimation in tomography, with proven minimax optimality and practical validation.

Findings

The estimator is minimax-optimal in Sobolev spaces.

The data-driven choice of the cut-off parameter is effective.

Simulation results show good practical performance.

Abstract

We study the non-parametric estimation of a multidimensional unknown density f in a tomography problem based on independent and identically distributed observations, whose common density is proportional to the Radon transform of f. We identify the underlying statistical inverse problem and use a spectral cut-off regularisation to deduce an estimator. A fully data-driven choice of the cut-off parameter m in R+ is proposed and studied. To discuss the bias-variance trade off, we consider Sobolev spaces and show the minimax-optimality of the spectral cut-off density estimator. In a simulation study, we illustrate a reasonable behaviour of the studied fully data-driven estimator.