We develop a methodology for conducting inference on extreme quantiles of unobserved individual heterogeneity (heterogeneous coefficients, heterogeneous treatment effects, etc.) in a panel data or meta-analysis setting. Inference in such settings is challenging: only noisy estimates of unobserved heterogeneity are available, and approximations based on the central limit theorem work poorly for extreme quantiles. For this situation, under weak assumptions we derive an extreme value theorem and an intermediate order theorem for noisy estimates and appropriate rate and moment conditions. Both theorems are then used to construct confidence intervals for extremal quantiles. The intervals are simple to construct and require no optimization. Inference based on the intermediate order theorem involves a novel self-normalized intermediate order theorem. In simulations, our extremal confidence intervals have favorable coverage properties in the tail. Our methodology is illustrated with an application to firm productivity in denser and less dense areas.

翻译：我们开发了一种方法，用于在面板数据或元分析设定中对未观测个体异质性（异质性系数、异质性处理效应等）的极端分位数进行推断。此类设定中的推断具有挑战性：仅能获得未观测异质性的含噪声估计，且基于中心极限定理的近似对极端分位数效果不佳。针对这一情况，在弱假设条件下，我们推导了含噪声估计的极值定理和中间阶定理，并给出了适当的速率与矩条件。这两个定理随后被用于构建极端分位数的置信区间。这些区间构建简便且无需优化。基于中间阶定理的推断涉及一种新颖的自归一化中间阶定理。在模拟中，我们的极端置信区间在尾部具有令人满意的覆盖性质。该方法通过一项关于密集与稀疏地区企业生产率的应用得到展示。