This paper studies an asymptotic framework for conducting inference onparameters of the form $phi(heta_0)$, where $phi$ is a known directionallydifferentiable function and $heta_0$ is estimated by $hat heta_n$. Inthese settings, the asymptotic distribution of the plug-in estimator $phi(hatheta_n)$ 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 $hat heta_n$ 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.
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机译：本文研究了一种渐进框架，用于对形式为$ phi（ theta_0）$的参数进行推理，其中$ phi $是已知的方向可微函数，而$ theta_0 $由$ hat theta_n $估计。在这些设置中，可以使用现有的Delta方法扩展轻松地得出插件估算器$ phi（ hat theta_n）$的渐近分布。但是，我们表明，“标准”引导程序仅在过于严格的条件下才是一致的；特别是，我们确定，只要$ hat theta_n $的限制分布是高斯分布，$ phi $的可区分性是引导程序一致性的必要和充分条件。提出了另一种重新采样方案，该方案在自举失败时仍保持一致，并显示为$ phi $的方向导数提供了局部大小控制约束。我们通过开发一个检验希尔伯特空间值参数是否属于凸集的方法来说明我们的结果的效用-该设置包括矩不等式问题和某些形状限制测试作为特殊情况。
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