Inference on Directionally Differentiable Functions

TL;DR

This paper develops a framework for inference on parameters transformed by directionally differentiable functions, highlighting bootstrap limitations and proposing an alternative resampling method with practical applications in shape restrictions and inequality testing.

Contribution

It establishes the necessity of differentiability for bootstrap consistency and introduces a new resampling scheme effective when bootstrap fails.

Findings

Bootstrap is consistent only under differentiability of the transformation.

An alternative resampling method remains valid when bootstrap fails.

Application to tests involving convex sets and moment inequalities.

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.