Quadratic functional estimation from observations with multiplicative measurement error

TL;DR

This paper develops a fully data-driven method for estimating quadratic functionals of a density from observations with multiplicative measurement error, achieving near-optimal convergence rates.

Contribution

It introduces a novel spectral cut-off estimator based on Mellin transform inversion and a data-driven parameter choice method, extending nonparametric quadratic functional estimation to multiplicative error models.

Findings

Estimator attains oracle-like convergence rates up to logarithmic factors.

Convergence rates are derived under classical smoothness assumptions.

Simulation studies validate the theoretical results.

Abstract

We consider the nonparametric estimation of the value of a quadratic functional evaluated at the density of a strictly positive random variable $X$ based on an iid. sample from an observation $Y$ of $X$ corrupted by an independent multiplicative error $U$. Quadratic functionals of the density covered are the $\mathbb{L}^2$-norm of the density and its derivatives or the survival function. We construct a fully data-driven estimator when the error density is known. The plug-in estimator is based on a density estimation combining the estimation of the Mellin transform of the $Y$ density and a spectral cut-off regularized inversion of the Mellin transform of the error density. The main issue is the data-driven choice of the cut-off parameter using a Goldenshluger-Lepski-method. We discuss conditions under which the fully data-driven estimator attains oracle-rates up to logarithmic deteriorations. We compute convergence rates under classical smoothness assumptions and illustrate them by a simulation study.