Nonparametric estimation of the conditional mode with errors-in-variables: Strong consistency for mixing processes

Having two variables, an explanatory one (X^o) and a response one (Y^o), linked by the classical relation Y^o = g(X^o) + ε, we want to estimate the function g(.) without any parametric assumption. However, in a lot of situations, the variables are not measured directly but through their proxies X = X^o + ε and Y = Y^o + η where ε and η are the measurements errors. We propose here a new method for estimating the function g(.) in such a context. Our estimator is based on deconvoluted kernels. Uniform convergence is established for strongly mixing stochastic processes. Some simulations show that our estimator is tractable and performs relatively well in practice.