  The Fisher-Rao distance between two probability distributions of a
statistical model is defined as the Riemannian geodesic distance induced by the
Fisher information metric. In order to calculate the Fisher-Rao distance in
closed-form, we need (1) to elicit a formula for the Fisher-Rao geodesics, and
(2) to integrate the Fisher length element along those geodesics. We consider
several numerically robust approximation and bounding techniques for the
Fisher-Rao distances: First, we report generic upper bounds on Fisher-Rao
distances based on closed-form 1D Fisher-Rao distances of submodels. Second, we
describe several generic approximation schemes depending on whether the
Fisher-Rao geodesics or pregeodesics are available in closed-form or not. In
particular, we obtain a generic method to guarantee an arbitrarily small
additive error on the approximation provided that Fisher-Rao pregeodesics and
tight lower and upper bounds are available. Third, we consider the case of
Fisher metrics being Hessian metrics, and report generic tight upper bounds on
the Fisher-Rao distances using techniques of information geometry.
Uniparametric and biparametric statistical models always have Fisher Hessian
metrics, and in general a simple test allows to check whether the Fisher
information matrix yields a Hessian metric or not. Fourth, we consider
elliptical distribution families and show how to apply the above techniques to
these models. We also propose two new distances based either on the Fisher-Rao
lengths of curves serving as proxies of Fisher-Rao geodesics, or based on the
Birkhoff/Hilbert projective cone distance. Last, we consider an alternative
group-theoretic approach for statistical transformation models based on the
notion of maximal invariant which yields insights on the structures of the
Fisher-Rao distance formula which may be used fruitfully in applications.
