A test of homogeneity of distributions when observations are subject to measurement errors.

When the observed data are contaminated with errors, the standard two-sample testing approaches that ignore measurement errors may produce misleading results, including a higher type-I error rate than the nominal level. To tackle this inconsistency, a nonparametric test is proposed for testing equality of two distributions when the observed contaminated data follow the classical additive measurement error model. The proposed test takes into account the presence of errors in the observed data, and the test statistic is defined in terms of the (deconvoluted) characteristic functions of the latent variables. Proposed method is applicable to a wide range of scenarios as no parametric restrictions are imposed either on the distribution of the underlying latent variables or on the distribution of the measurement errors. Asymptotic null distribution of the test statistic is derived, which is given by an integral of a squared Gaussian process with a complicated covariance structure. For data-based calibration of the test, a new nonparametric Bootstrap method is developed under the two-sample measurement error framework and its validity is established. Finite sample performance of the proposed test is investigated through simulation studies, and the results show superior performance of the proposed method than the standard tests that exhibit inconsistent behavior. Finally, the proposed method was applied to real data sets from the National Health and Nutrition Examination Survey. An R package MEtest is available through CRAN.