Bounds and expansions for Fisher information when the moments are known

An expansion of the score statistic as a polynomial in the random variable can be used to provide a nondecreasing sequence of lower bounds for the Fisher information when the moments are known as functions of the parameter. If the random variable is asymptotically normal, the first three bounds provide the first three terms in the asymptotic expansion for the Fisher information. This leads to a definition of efficiency which may be used to advantage with small sample sizes, providing lower bounds for the efficiency of any statistic whose moments are known but whose exact distribution is intractable.