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

Abstract

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

Doubly Robust Inference when Combining Probability and Non-Probability Samples with High Dimensional Data

“…Let p A ={ p A, i : i =1,…, N } with be the inclusion probabilities for the conditional Poisson sampling design. Given p A , it is possible to find π A ={ π A, i : i =1,…, N } with such that the conditional Poisson sampling design with p A is asymptotically equivalent to the Poisson sampling design with the inclusion probabilities π A (Hájek, 1964; Conti, ). Therefore, for a high entropy design, to apply our theoretical results, we check the conditions for the corresponding inclusion probability π A under Poisson sampling.…”

Inference for Quantiles of a Finite Population: Asymptotic versus Resampling Results

“…Because of its properties, we consider here the classical Hájek estimator of F N : which is a proper distribution function. Its main asymptotic properties are summarized in proposition , proved in Conti ().…”

“…Sitter & Wu () showed that the Woodruff intervals perform well even in moderate to extreme tail regions of the distribution function. In fact, as shown in Conti (), the term ( A − 1) p (1 − p ) is essentially the asymptotic variance of the Hájek estimator when y = Q ( p ).…”

“…which is a proper distribution function. Its main asymptotic properties are summarized in proposition 1, proved in Conti (2014).…”

“…Apart from the articles by Rosén () and Sen (), which are confined to s.r.s., in our knowledge, the only paper devoted to the estimation of the whole p.d.f. for general sampling designs is Conti (), where a ‘finite‐population version’ of the empirical process is considered, and its weak convergence to a Gaussian process is proved.…”