Inference on Directionally Differentiable Functions

Abstract: This paper studies an asymptotic framework for conducting inference on parameters of the form $\phi(\theta_0)$, where $\phi$ is a known directionally differentiable function and $\theta_0$ is estimated by $\hat \theta_n$. In these settings, the asymptotic distribution of the plug-in estimator $\phi(\hat \theta_n)$ can be readily derived employing existing extensions to the Delta method. We show, however, that the "standard" bootstrap is only consistent under overly stringent conditions -- in particular we establish that differentiability of $\phi$ is a necessary and sufficient condition for bootstrap consistency whenever the limiting distribution of $\hat \theta_n$ is Gaussian. An alternative resampling scheme is proposed which remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of $\phi$. We illustrate the utility of our results by developing a test of whether a Hilbert space valued parameter belongs to a convex set -- a setting that includes moment inequality problems and certain tests of shape restrictions as special cases.