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Fisher information is a measure of the best precision with which a parameter can be estimated from statistical data. It can also be defined for a continuous random variable without reference to any parameters, in which case it has a physically compelling interpretation of representing the highest precision with which the first cumulant of the random variable, i.e., its mean, can be estimated from its statistical realizations. We construct a complete hierarchy of information measures that determine the best precision with which all of the cumulants of a random variable - and thus its complete probability distribution - can be estimated from its statistical realizations. Several properties of these information measures and their generating functions are discussed.

@article{Prasad2004638, abstract = {Fisher information is a measure of the best precision with which a parameter can be estimated from statistical data. It can also be defined for a continuous random variable without reference to any parameters, in which case it has a physically compelling interpretation of representing the highest precision with which the first cumulant of the random variable, i.e., its mean, can be estimated from its statistical realizations. We construct a complete hierarchy of information measures that determine the best precision with which all of the cumulants of a random variable - and thus its complete probability distribution - can be estimated from its statistical realizations. Several properties of these information measures and their generating functions are discussed.}, art_number = {638-642}, author = {Prasad, S. and Menicucci, N.C.}, date-added = {2019-03-18 14:39:26 +1100}, date-modified = {2019-03-19 16:35:04 +1100}, doi = {10.1109/TIT.2004.825034}, journal = {IEEE Transactions on Information Theory}, number = {4}, title = {Fisher information with respect to cumulants}, url_link = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-1942454268&doi=10.1109%2fTIT.2004.825034&partnerID=40&md5=c63aa77abedc12bcdca35049231d1f35}, volume = {50}, year = {2004}, Bdsk-Url-1 = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-1942454268&doi=10.1109%2fTIT.2004.825034&partnerID=40&md5=c63aa77abedc12bcdca35049231d1f35}, Bdsk-Url-2 = {https://doi.org/10.1109/TIT.2004.825034}}

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