  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.
