## Differential Geometry in Statistical Inference |

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### Common terms and phrases

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

### Popular passages

Page 238 - Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, J.

Page 93 - Second order efficiency of minimum contrast estimators in a curved exponential family.

Page 240 - Convexity properties of entropy functions and analysis of diversity. In Inequalities in Statistics and Probability, IMS Lecture Notes, 5, 68-77.

Page 90 - Acknowledgements The author would like to express his sincere gratitude to Dr.

Page 161 - A note on the inverse Gaussian distribution function. J. Amer. Statist. Assoc. 63, 1514-1516.

Page 91 - Estimation of a structural parameter in the presence of a large number of nuisance parameters, Biometnka 71.

Page 21 - Wiener processes, and so on, have so far been studied in detail, it seems rather strange that only a few theories have been proposed concerning properties of a family itself of distributions. Here, by the properties of a family we mean such geometric relations as mutual distances, flatness or curvature of the family, etc.

Page 94 - Nagaoka, H. and Amari, S. (1982). Differential geometry of smooth families of probability distributions, METR 82-7, Univ. Tokyo.

Page 14 - This paper was prepared with support from the National Science Foundation under Grant No.

Page 240 - Radhakrishna and Nayak, TK (1985). Cross entropy, dissimilarity measures and characterizations of quadratic entropy.