  We consider a circular deconvolution problem, in which the density $f$ of a
circular random variable $X$ must be estimated nonparametrically based on an
i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement
error is supposed to be independent of $X$. The objective of this work was to
construct a fully data-driven estimation procedure when the error density
$\varphi$ is unknown. We assume that in addition to the i.i.d. sample from $Y$,
we have at our disposal an additional i.i.d. sample drawn independently from
the error distribution. We first develop a minimax theory in terms of both
sample sizes. We propose an orthogonal series estimator attaining the minimax
rates but requiring optimal choice of a dimension parameter depending on
certain characteristics of $f$ and $\varphi$, which are not known in practice.
The main issue addressed in this work is the adaptive choice of this dimension
parameter using a model selection approach. In a first step, we develop a
penalized minimum contrast estimator assuming that the error density is known.
We show that this partially adaptive estimator can attain the lower risk bound
up to a constant in both sample sizes $n$ and $m$. Finally, by randomizing the
penalty and the collection of models, we modify the estimator such that it no
longer requires any previous knowledge of the error distribution. Even when
dispensing with any hypotheses on $\varphi$, this fully data-driven estimator
still preserves minimax optimality in almost the same cases as the partially
adaptive estimator. We illustrate our results by computing minimal rates under
classical smoothness assumptions.
