Doubly Robust Estimation of the Finite Population Distribution Function Using Nonprobability Samples

In official statistics, quantiles and other distributional indicators are essential for monitoring income distribution and inequality and for producing policy-relevant benchmarks such as medians, deciles, and thresholds. Correspondingly, extensive methods exist for complex survey designs; by contrast, practical tools for nonprobability samples remain limited despite their growing use. We address this gap by developing estimators of the finite population distribution function within a data integration framework that combines nonprobability and probability samples. Specifically, we propose an inverse probability weighted estimator, a regression estimator, and a doubly robust estimator, and we show that the doubly robust estimator is asymptotically unbiased for the finite distribution function when either the propensity score model or the outcome regression model is correctly specified. Building upon the estimated finite distribution function, we derive quantile estimators and construct Woodruff confidence intervals, employing bootstrap variance estimation specifically tailored to this framework. Simulation studies based on a synthetic population and the 2023 Korean Survey of Household Finances and Living Conditions indicate that the proposed estimators perform well in practice, supporting their usefulness for official statistical production.