Abstract: In the analysis of complex survey data, it is often important to estimate distribution functions or quantiles associated with a given variable x. Estimation of these values is known to be problematic if the variable x is measured with error. Previous work with this problem generally has used the assumption that the measurement errors, or transformations thereof, have a normal distribution. This paper uses small-error approximations to develop adjusted estimators of distribution functions or quantiles for cases in which measurement errors are nonnormal. These approximations also lead to some relatively simple diagnostics that indicate the extent to which customary sample survey distribution function estimators are sensitive to: (1) varying magnitudes of measurement error; and (2) the approximate shape of the distribution of the true x values. Some of the proposed diagnostics require identification information, e.g., estimates of the measurement error variance, but do not necessarily require direct access to individual-level replicate observations. The proposed methods are applied to data from the U.S. Third National Health and Nutrition Examination Survey (NHANES III).
Key words and phrases: Convolution, misclassification, nonnormal distribution, percentile, prevalence rate, response error, sensitivity analysis, small-error approximation, stratified multistage sample survey, superpopulation model, Third National Health and Nutrition Examination Survey (NHANES III).