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

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.

Empirical partially Bayes multiple testing and compound $χ^2$ decisions

“…The key technical argument for the proof (in Supplement E.2) hinges on properties of the Mellin transform [Epstein, 1948, Butzer and Stefan, 1999, Brenner Miguel et al, 2021.…”

“…As mentioned in the main text, our proof of Lemma 10 builds upon Mellin transform analysis. For conciseness we do not provide the necessary background and definitions here and instead refer to Brenner Miguel et al [2021]; our notation in the proof matches the notation therein. We do however provide references for any mathematical property of the Mellin transform that we invoke.…”