Stochastic interventions are a powerful tool to define parameters that measure the causal effect of a realistic intervention that intends to alter the population distribution of an exposure. In this paper we follow the approach described in D\'iaz and van der Laan (2011) to define and estimate the effect of an intervention that is expected to cause a truncation in the population distribution of the exposure. The observed data parameter that identifies the causal parameter of interest is established, as well as its efficient influence function under the non parametric model. Inverse probability of treatment weighted (IPTW), augmented IPTW and targeted minimum loss based estimators (TMLE) are proposed, their consistency and efficiency properties are determined. An extension to longitudinal data structures is presented and its use is demonstrated with a real data example.
Díaz, Iván and van der Laan, Mark J., "Assessing the Causal Effect of Policies: An Approach Based on Stochastic Interventions" (October 2012). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 298.