Frangakis et al. (2015) proposed a numerical method for computing the efficient influence function of a parameter in a nonparametric model at a specified distribution and observation (provided such an influence function exists). Their approach is based on the assumption that the efficient influence function is given by the directional derivative of the target parameter mapping in the direction of a perturbation of the data distribution defined as the convex line from the data distribution to a pointmass at the observation. In our discussion paper Luedtke et al. (2015) we propose a regularization of this procedure and establish the validity of this method in great generality. In this article we propose a generalization of the latter regularized numerical delta method for computing the efficient influence function for general statistical models, and formally establish its validity under appropriate regularity conditions. Our proposed method consists of applying the regularized numerical delta-method for nonparametrically-defined target parameters proposed in Luedtke et al. 2015 to the nonparametrically-defined maximum likelihood mapping that maps a data distribution (normally the empirical distribution) into its Kullback-Leibler projection onto the model. This method formalizes the notion that an algorithm for computing a maximum likelihood estimator also yields an algorithm for computing the efficient influence function at a user-supplied data distribution. We generalize this method to a minimum loss-based mapping. We also show how the method extends to compute the higher-order efficient influence function at an observation pair for higher-order pathwise differentiable target parameters. Finally, we propose a new method for computing the efficient influence function as a whole curve by applying the maximum likelihood mapping to a perturbation of the data distribution with score equal to an initial gradient of the pathwise derivative. We demonstrate each method with a variety of examples.
van der Laan, Mark J.; Carone, Marco; and Luedtke, Alexander R., "Computerizing Efficient Estimation of a Pathwise Differentiable Target Parameter" (July 2015). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 340.