Fisher Information With Respect to Cumulants

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.