Abstract
Curved exponential families are so general objects that they seem to have no interesting universal properties. However Abram Kagan [1] discovered in 1985 a remarkable inequality on their Fisher information. This note gives a modern presentation of this result and examples, comparing in particular noncentral and central Wishart distributions.
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References
Kagan, A.: An information property of exponential families. Teor. Veroyatnost. i Primenen. 30(4), 783–786 (1985), Theory Probab. Appl. 30(4), 831–835 (1986)
Letac, G., Massam, H.: The noncentral Wishart as an exponential family and its moments. J. Multivariate Anal. 99, 1393–1417 (2008)
Letac, G., Massam, H.: The Laplace transform \((\det s)^{-p}\exp \rm trace\, ( s^{-1})\) and the existence of the non-central Wishart distributions. J. Multivariate Anal. 163, 96–110 (2018)
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Letac, G. (2021). The Fisher Information of Curved Exponential Families and the Elegant Kagan Inequality. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_36
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DOI: https://doi.org/10.1007/978-3-030-80209-7_36
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