Nonparametric curve estimation in measurement error problems with conditionally heteroscedastic variances

Abstract

We consider the problem of estimating the density 𝑓𝑋 of a latent variable X using replicated observations contaminated by additive errors. Unlike the classical setting, these errors depend on X, which makes standard Fourier deconvolution techniques inappropriate, and we suggest instead a basis expansion approach. Using a basis of Legendre polynomials, we show that it is possible to express the unknown coefficients of the expansion in terms of invertible equations of moments of the observed data. We deduce a nonparametric estimator of 𝑓𝑋 which attains optimal minimax convergence rates in the case where the errors are conditionally Gaussian. To implement our density estimator in practice, ..