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a-connections action affine connection Amari ancillary ancillary family approximation associated assume asymptotic Barndorff-Nielsen basis calculated called coefficient components conditional connection consider consistent constant construct coordinate system correspondence curvature curved exponential family defined denotes density depend derivative determined differential direction discussed distribution efficient elements equal equation estimator example exists expansion expected expression field Fisher information fixed foliation function further Gaussian geodesic geometry given gives Hence identity interest introduced invariant inverse linear loss manifold mapping matrix maximum likelihood measure metric minimal mixture natural normal Note observed obtained orthogonal parallel parameter particular probability problem properties relation represents respect Riemannian sample satisfies statistical statistical manifold statistical model structure submanifold sufficient symmetric tangent space tangent vector tensor Theorem theory third-order tion transformation models vector
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Page 90 - Acknowledgements The author would like to express his sincere gratitude to Dr.
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Page 14 - This paper was prepared with support from the National Science Foundation under Grant No.