= Functional statistic
{title2=$T(F_n)$}
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>, then
$$
T(F_n)-T(F)=\frac1n\sum_i\operatorname{IF}(X_i;T,F)+o_p(n^{-1/2}).
$$
A square-integrable <influence function> with mean zero consequently gives <asymptotic variance> $n^{-1}\int\operatorname{IF}(x;T,F)^2\,dF(x)$ by the <central limit theorem>.
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