On the Estimation of the Distribution Function of a Finite Population Under High Entropy Sampling Designs, with Applications

The estimation of the distribution function of a population is an important problem in sampling finite populations. The existing literature focuses on the problem of estimating the population distribution function (p.f.d.) at a single point, or at a finite number of points. In this paper the main interest consists in estimating the whole p.d.f.. In many respects, the starting point is close to classical nonparametric statistics, although the approach to inference is based on sampling design. It is shown here that the Hajek estimator of the p.d.f., if properly centered and scaled, converges weakly to a Gaussian process with covariance kernel proportional to that of a Brownian bridge. The proportionality factor essentially depends on the sample design. Applications to (i) construction of a confidence band for the p.d.f., (ii) comparison of the p.d.f.s of two populations, and (iii) testing for independence of two characters are provided.