Let
![$\Theta $](/math_tex/f7cb07c8f1be682cae5dddbdc214557d82.gif)
be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space
![$E^s$](/math_tex/b4db4ab34070c4cca9a857cdc814502782.gif)
, and let
![$\gamma$](/math_tex/11c596de17c342edeed29f489aa4b27482.gif)
be a smooth map of
![$\Theta$](/math_tex/b35e24d8a08c0ab01195f2ad2a78fab782.gif)
into a Riemannian manifold Λ. An unknown state
![$\theta \in \Theta$](/math_tex/1694fa79d012a58f8baabdf8e497421682.gif)
is observed via X =
![$\theta + \epsilon \xi$](/math_tex/9aa66c70e60652e33fcca0105132683782.gif)
, where
![$\epsilon$](/math_tex/7ccca27b5ccc533a2dd72dc6fa28ed8482.gif)
> 0 is a small parameter and
![$\xi$](/math_tex/85e60dfc14844168fd12baa5bfd2517d82.gif)
is a white Gaussian noise. For a given smooth prior
![$\lambda$](/math_tex/fd8be73b54f5436a5cd2e73ba9b6bfa982.gif)
on
![$\Theta$](/math_tex/b35e24d8a08c0ab01195f2ad2a78fab782.gif)
and smooth estimators
![$g(X)$](/math_tex/889ff6fa193255b40c2616db0e8ec14f82.gif)
of the map
![$\gamma$](/math_tex/11c596de17c342edeed29f489aa4b27482.gif)
we derive a secondorder asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of
the underlying spaces
![$\Theta$](/math_tex/b35e24d8a08c0ab01195f2ad2a78fab782.gif)
and
![$\Lambda$](/math_tex/b23332f99af850a48831f80dbf681ed682.gif)
, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of
![$\gamma$](/math_tex/11c596de17c342edeed29f489aa4b27482.gif)
is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.