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：我们提出了一种方法，用于在面板数据或元分析背景下对未观测个体异质性（如异质性系数、异质性处理效应等）的极端分位数进行推断。此类背景下的推断具有挑战性：仅能获得未观测异质性的含噪声估计，而基于中心极限定理的近似方法在极端分位数处效果不佳。针对这一情况，我们在弱假设下推导了含噪声估计的极值定理和中阶定理，并给出了合适的速率与矩条件。利用这两个定理，我们构建了极端分位数的置信区间。这些区间构造简单且无需优化。基于中阶定理的推断涉及一种新型的自归一化中阶定理。在模拟实验中，我们的极端置信区间在尾部具有良好的覆盖性质。该方法通过一个关于高密度与低密度地区企业生产率的应用实例加以说明。