SIER Working Paper Series

This paper is concerned with inference on the cumulative distribution function (cdf) FX ∗ in the classical measurement error model X = X ∗ + . We consider the case where the density of the measurement error is unknown and estimated by repeated measurements, and show validity of a bootstrap approximation for the distribution of the deviation in the sup-norm between the deconvolution cdf estimator and FX ∗ . We allow the density of to be ordinary or super smooth. We also provide several theoretical results on the bootstrap and asymptotic Gumbel approximations of the sup-norm deviation for the case where the density of is known. Our approximation results are applicable to various contexts, such as confidence bands for FX ∗ and its quantiles, and for performing various cdf-based tests such as goodness-of-fit tests for parametric models of X ∗ , two sample homogeneity tests, and tests for stochastic dominance. Simulation and real data examples illustrate satisfactory performance of the proposed methods.