  The success of deep learning has inspired recent interests in applying neural
networks in statistical inference. In this paper, we investigate the use of
deep neural networks for nonparametric regression with measurement errors. We
propose an efficient neural network design for estimating measurement error
models, in which we use a fully connected feed-forward neural network (FNN) to
approximate the regression function $f(x)$, a normalizing flow to approximate
the prior distribution of $X$, and an inference network to approximate the
posterior distribution of $X$. Our method utilizes recent advances in
variational inference for deep neural networks, such as the importance weight
autoencoder, doubly reparametrized gradient estimator, and non-linear
independent components estimation. We conduct an extensive numerical study to
compare the neural network approach with classical nonparametric methods and
observe that the neural network approach is more flexible in accommodating
different classes of regression functions and performs superior or comparable
to the best available method in nearly all settings.
