A differential geometric framework to construct an asymptotically unbiased
estimator of a function of a parameter is presented. The derived estimator
asymptotically coincides with the uniformly minimum variance unbiased
estimator, if a complete sufficient statistic exists. The framework is based on
the maximum a posteriori estimation, where the prior is chosen such that the
estimator is unbiased. The framework is demonstrated for the second-order
asymptotic unbiasedness (unbiased up to O(nβ1) for a sample of size n).
The condition of the asymptotic unbiasedness leads the choice of the prior such
that the departure from a kind of harmonicity of the estimand is canceled out
at each point of the model manifold. For a given estimand, the prior is given
as an integral. On the other hand, for a given prior, we can address the bias
of what estimator can be reduced by solving an elliptic partial differential
equation. A family of invariant priors, which generalizes the Jeffreys prior,
is mentioned as a specific example. Some illustrative examples of applications
of the proposed framework are provided.Comment: 28 pages, 2 figure