We consider a situation where a distribution is being estimated by the empirical distribution of noisy measurements. The measurements errors are allowed to be heteroskedastic and their variance may depend on the realization of the underlying random variable. We use an asymptotic embedding where the noise shrinks with the sample size to calculate the leading bias arising from the presence of noise. Conditions are obtained under which this bias is asymptotically non-negligible. Analytical and jackknife corrections for the empirical distribution are derived that recenter the limit distribution and yield confidence intervals with correct coverage in large samples. Similar adjustments are presented for nonparametric estimators of the density and quantile function. Our approach can be connected to corrections for selection bias and shrinkage estimation. Simulation results confirm the much improved sampling behavior of the corrected estimators. An empirical application to the estimation of a stochastic-frontier model is also provided.