Nonparametric density estimation for stationary processes under multiplicative measurement errors

This paper focuses on estimating the invariant density function fX of the strongly mixing stationary process Xₜ in the multiplicative measurement errors model Yₜ = Xₜ Uₜ, where Uₜ is also a strongly mixing stationary process. We propose a novel approach to handle non-independent data, typical in real-world scenarios. For instance, data collected from various groups may exhibit interdependencies within each group, resembling data generated from m-dependent stationary processes, a subset of stationary processes. This study extends the applicability of the model Yₜ = Xₜ Uₜ to diverse scientific domains dealing with complex dependent data. The paper outlines our estimation techniques, discusses convergence rates, establishes a lower bound on the minimax risk, and demonstrates the asymptotic normality of the estimator for fX under smooth error distributions. Through examples and simulations, we showcase the efficacy of our estimator. The paper concludes by providing proofs for the presented theoretical results. v