Multiscale density estimation with errors in variables

We present an adaptive method for density estimation when the observations X=(X1,…,Xn) are contaminated by additive errors Y=X+Z. The error distribution is not specified by the model but is estimated using repeated measurements (Yi)i=1,2,3 of X. In this setting, we propose a wavelet method for density estimation which adapts both to the degree of ill posedness of the problem (smoothness of the error distribution) and to the regularity of the target density. Our method is implemented in the Fourier domain via a square-root transformation of the empirical characteristic function and yields fast translation invariant non-linear wavelet approximations with data-driven choices of fine tuning parameters. When the variable X is observed without errors our method provides a natural implementation of direct density estimation in the Meyer wavelet basis. We illustrate the adaptiveness properties of our estimator with a range of finite sample examples drawn from population with smooth and less smooth density functions.