Functional statistic ($T(F_n)$)

A functional statistic evaluates a statistical functional $T$ at the empirical distribution $F_n=n^{-1}{\sum }_i{\delta }_{X_i}$. The map $T$ acts on distributions, whereas $T(F_n)$ is a statistic. If $T$ has an asymptotic linear representation, thenA square-integrable influence function with mean zero consequently gives asymptotic variance $n^{-1}\int \mathrm{IF}(x;T,F{)}^{2}dF(x)$ by the central limit theorem.