Often semiparametric estimators are asymptotically equivalent to a sample average. The object being averaged is referred to as the inﬂuence function. The inﬂuence function is useful in formulating primitive regularity conditions for asymptotic normality, in efficiency comparions, for bias reduction, and for analyzing robustness. We show that the inﬂuence function of a semiparametric estimator can be calculated as the limit of the Gateaux derivative of a parameter with respect to a smooth deviation as the deviation approaches a point mass. We also consider high level and primitive regularity conditions for validity of the inﬂuence function calculation. The conditions involve Frechet differentiability, nonparametric convergence rates, stochastic equicontinuity, and small bias conditions. We apply these results to examples.
7 August 2015
Working Paper (CWP44/15)