Influence Function: Local Robustness and Efficiency

We propose a direct approach to calculating influence functions based on the concept of functional derivatives. The relative simplicity of our direct method is demonstrated through well-known examples. Using influence functions as a key device, we examine the connection and difference between local robustness and efficiency in both joint and sequential identification/estimation procedures. We show that the joint procedure is associated with efficiency, while the sequential procedure is linked to local robustness. Furthermore, we provide conditions that are theoretically verifiable and empirically testable on when efficient and locally robust estimation for the parameter of interest in a semiparametric model can be achieved simultaneously. In addition, we present straightforward conditions for an adaptive procedure in the presence of nuisance parameters.