Coarsening-adjusted estimators for finite-population indicators

Self-reported numerical variables in sample surveys are frequently subject to coarsening, as respondents tend to report rounded or heaped values rather than the exact underlying quantity. While the statistical literature offers various model-based solutions, official statistics and public health surveillance routinely require design-based inference on finite-population indicators under complex sampling designs. This paper introduces a general framework that bridges this gap by treating the observed response as a coarsened manifestation of a latent variable, modeling reporting regimes of increasing coarseness jointly with the latent distribution via a survey-weighted pseudo-likelihood approach. The fitted model is used to generate posterior predictive replicates of the latent values, to which standard design-based estimators are applied, formally propagating the additional uncertainty through an explicit variance decomposition. Simulation studies assess the finite-sample performance and robustness of the proposed method under various misspecification and sampling scenarios. Finally, its practical relevance is demonstrated through an application to behavioural data from the Italian PASSI surveillance system, illustrating how coarsening affects different functionals, such as means, quantiles, and threshold-based prevalence indicators in heterogeneous ways.