Spectral cut-off regularisation for density estimation under multiplicative measurement errors

TL;DR

This paper introduces a spectral cut-off regularisation method for non-parametric density estimation on positive real numbers, effectively handling multiplicative measurement errors through Mellin transform techniques and adaptive model selection.

Contribution

It proposes a fully data-driven, minimax-optimal density estimator using Mellin transform regularisation and introduces Mellin-Sobolev spaces to characterize regularity.

Findings

The estimator achieves minimax-optimal rates over Mellin-Sobolev spaces.

The method adapts to unknown regularity of the density.

The approach effectively manages bias-variance trade-off in multiplicative error models.

Abstract

We study the non-parametric estimation of an unknown density f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed fully data driven procedure is based on the estimation of the Mellin transform of the density f , a regularisation of the inverse of the Mellin transform by a spectral cut-off and a data-driven model selection in order to deal with the upcoming bias-variance trade-off. We introduce and discuss further Mellin-Sobolev spaces which characterize the regularity of the unknown density f through the decay of its Mellin transform. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the data-driven density estimator and hence its adaptivity.