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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. |
