English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper studies an asymptotic framework for conducting inference onparameters of the form $\phi(\theta_0)$, where $\phi$ is a known directionallydifferentiable function and $\theta_0$ is estimated by $\hat \theta_n$. Inthese settings, the asymptotic distribution of the plug-in estimator $\phi(\hat\theta_n)$ can be readily derived employing existing extensions to the Deltamethod. We show, however, that the "standard" bootstrap is only consistentunder overly stringent conditions -- in particular we establish thatdifferentiability of $\phi$ is a necessary and sufficient condition forbootstrap consistency whenever the limiting distribution of $\hat \theta_n$ isGaussian. An alternative resampling scheme is proposed which remains consistentwhen the bootstrap fails, and is shown to provide local size control underrestrictions on the directional derivative of $\phi$. We illustrate the utilityof our results by developing a test of whether a Hilbert space valued parameterbelongs to a convex set -- a setting that includes moment inequality problemsand certain tests of shape restrictions as special cases.

Inference on Directionally Differentiable Functions