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.
翻译:我们考虑的是随机变量的分布是通过该变量的噪音测量的实证分布来估计的。这是教师增值模型和其他固定效应模型中常见的做法,例如用于小组数据的教师增值模型和其他固定效应模型。我们使用一个无症状嵌入器,噪音随着抽样规模的缩小而缩小,以计算由于噪音的存在而导致的经验分布中的主要偏差。经验量函数中的主要偏差也同样得到。这些计算在文献中是新颖的,只得出了平均和差异等顺利功能的结果。我们提供了分析和jacknife修正,以更新限制分布和产生信任间隔,在大样本中进行正确覆盖。我们的方法可以与选择偏差和缩小估计的纠正相连接,并与演化法作对比。模拟结果证实了经更正的估量者大大改进的抽样行为。我们同样提供了关于偏离一种价格法的偏差的经验性说明。