This paper considers estimators of the distribution function of a variable over a finite population, when a sample of units is available and values of a related auxiliary variable are known for the whole population. Theory is offered for several estimators which rely on nonparametric regression, and for calibration estimators which require a parametric model.
"Estimators of the Finite Population Distribution Function using Nonparametric Regression." Ann. Statist. 21 (3) 1452 - 1475, September, 1993. https://doi.org/10.1214/aos/1176349267