Density Deconvolution in the Circular Structural Model

We consider deconvolving bivariate irregular densities supported on the circumference of the unit circle. The errors are bivariate, and the observations are available on the plane. Assuming that the estimated density is smooth on the circle, we compute exact asymptotics of the minimax risks and develop asymptotically optimal estimators for the case of normal errors. The proposed estimators are automatically sharp minimax adaptive over a wide collection of smoothness classes. It is shown that the same rates of convergence hold for a variety of different types of error distributions. The interesting feature of the problem is that the optimal rates of convergence do not depend on the error distribution and are determined essentially by the problem geometry.