We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of that variable. This is common practice in, for example, teacher value-added models and other fixed-effect models for panel data. We use an asymptotic embedding where the noise shrinks with the sample size to calculate the leading bias in the empirical distribution arising from the presence of noise. The leading bias in the empirical quantile function is equally obtained. These calculations are new in the literature, where only results on smooth functionals such as the mean and variance have been derived. We provide both analytical and jackknife corrections that recenter the limit distribution and yield confidence intervals with correct coverage in large samples. Our approach can be connected to corrections for selection bias and shrinkage estimation and is to be contrasted with deconvolution. Simulation results confirm the much-improved sampling behavior of the corrected estimators. An empirical illustration on heterogeneity in deviations from the law of one price is equally provided.

我们考虑这样一种情况：随机变量的分布是通过该变量的噪声测量的经验分布来估计的。这是教师增值模型和其他面板数据固定效应模型中的常见做法。我们使用渐近嵌入，其中噪声随着样本大小而缩小，以计算由于噪声的存在而产生的经验分布中的主导偏差。同样获得经验分位数函数中的领先偏差。这些计算在文献中是新的，其中仅导出了均值和方差等平滑函数的结果。我们提供分析校正和折刀校正，使极限分布居中并产生置信区间，并在大样本中进行正确的覆盖。我们的方法可以与选择偏差和收缩估计的校正联系起来，并与反卷积进行对比。仿真结果证实了修正后的估计器的采样行为得到了很大改进。同样提供了关于偏离一价定律的异质性的实证说明。