  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.
