  Data in non-Euclidean spaces are commonly encountered in many fields of
Science and Engineering. For instance, in Robotics, attitude sensors capture
orientation which is an element of a Lie group. In the recent past, several
researchers have reported methods that take into account the geometry of Lie
Groups in designing parameter estimation algorithms in nonlinear spaces.
Maximum likelihood estimators (MLE) are quite commonly used for such tasks and
it is well known in the field of statistics that Stein's shrinkage estimators
dominate the MLE in a mean-squared sense assuming the observations are from a
normal population. In this paper, we present a novel shrinkage estimator for
data residing in Lie groups, specifically, abelian or compact Lie groups. The
key theoretical results presented in this paper are: (i) Stein's Lemma and its
proof for Lie groups and, (ii) proof of dominance of the proposed shrinkage
estimator over MLE for abelian and compact Lie groups. We present examples of
simulation studies of the dominance of the proposed shrinkage estimator and an
application of shrinkage estimation to multiple-robot localization.
