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The chain rule for derivatives of a function of a function is extended to afunction of a statistical functional, and applied to obtain approximations tothe cumulants, distribution and quantiles of functions of sample moments, andso to obtain third order confidence intervals and estimates of reduced bias forfunctions of moments. As an example we give the distribution of thestandardized skewness for a normal sample to magnitude $O(n^{-2})$, where $n$is the sample size.

The chain rule for functionals with applications to functions of moments