Inference on distribution functions under measurement error

This paper is concerned with inference on the cumulative distribution function (cdf) F_X^∗ in the classical measurement error model X=X^∗+ϵ. We consider the case where the density of the measurement error ϵ 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 F_X^∗ . We allow the density of ϵ 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 ϵ is known. Our approximation results are applicable to various contexts, such as confidence bands for F_X^∗ 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.