Linear functional estimation under multiplicative measurement error

Abstract

We study the non-parametric estimation of the value of a linear functional evaluated at an unknown density function f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed plug-in estimation procedure combines the estimation of the Mellin transform of the density f and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The attainable accuracy of the estimator is essentially determined by the decay of the upcoming Mellin transforms and the smoothness of the linear functional which we illustrate by different scenarios. As usual the choice of the cut-off parameter is crucial and we propose its data-driven selection inspired by the work of (Goldenshluger and Lepski Ann. Statist. 39 (2011) 1608–1632). By proving matching lower bounds we show that the plug-in estimator with optimally chosen cut-off parameter attains minimax-optimal rates of convergence over Mellin-Sobolev spaces. Furthermore the rate of convergence of the data-driven estimator features at most a deterioration by a logarithmic factor which is widely considered as an acceptable price for adaptation. In particular, our theory covers point-wise estimation of the density f, its derivative and Laplace transform, its associated survival and cumulative distribution function as well as the point-wise estimation of the mean residual life.