Inference on a distribution from noisy draws

Abstract

We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of the random variable. This is common practice in many settings, including the evaluation of teacher value-added and the assessment of firm efficiency through stochastic-frontier models. 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. 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. A similar adjustment is also presented for the quantile function. These corrections are non-parametric and easy to implement. 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 the estimation of a stochastic-frontier model is also provided.