Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error

In this paper, we study density estimation for mixed Euclidean and non-Euclidean variables that are subject to measurement errors. This problem is largely unexplored in statistics. We develop a new deconvolution density estimator and derive its finite-sample properties. We also derive its asymptotic properties including the rate of convergence in various modes and the asymptotic distribution. For the derivation, we apply Fourier analysis on topological groups, which has not been well used in statistics. We provide full practical details on the implementation of the estimator as well as several simulation studies and real data analysis.