  In decision theoretic estimation of parameters in Euclidean space
$\mathbb{R}^p$, the action space is chosen to be the convex closure of the
estimand space. In this paper, the concept has been extended to the estimation
of circular parameters of distributions having support as a circle, torus or
cylinder. As directional distributions are of curved nature, existing methods
for distributions with parameters taking values in $\mathbb{R}^p$ are not
immediately applicable here. A circle is the simplest one-dimensional
Riemannian manifold. We employ concepts of convexity, projection, etc., on
manifolds to develop sufficient conditions for inadmissibility of estimators
for circular parameters. Further invariance under a compact group of
transformations is introduced in the estimation problem and a complete class
theorem for equivariant estimators is derived. This extends the results of
Moors [J. Amer. Statist. Assoc. 76 (1981) 910-915] on $\mathbb{R}^p$ to
circles. The findings are of special interest to the case when a circular
parameter is truncated. The results are implemented to a wide range of
directional distributions to obtain improved estimators of circular parameters.
