Quadratic functional estimation from observations with multiplicative measurement error

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.