The normal approximation for semiparametric averaged derivatives

With the same normalization as that for standard parametric statistics, and centered at a parameter of interest, many semiparametric estimates based on n observations have been shown to be root-n-consistent and asymptotically normal. In the context of semiparametric averaged derivative estimates, we go further by showing that the rate of convergence of the finite-sample distribution to the normal limit distribution can equal that of standard parametric statistics.