Informative designs in survey sampling: an asymptotic approach

The present dissertation focuses on informative designs, sampling methods in which inclusion probabilities depend on the variable of interest. We first study inverse sampling, a special informative design introduced for studying rare populations. In its simplest formulation, it is described as sequential sampling, where units are drawn until a fixed number of rare units have been sampled. Next, we study the estimation of the distribution function under an unknown informative design. Particular emphasis is given to the asymptotic approach, where both the population and sample size increase. We begin by presenting two counting processes related to the inverse selection scheme with equal probabilities. Then, we review the classical large-sample properties of simple random sampling and unequal probability sampling with replacement. This theory, together with limit theorems for randomly indexed sequences of random variables, forms the background for developing asymptotic results for unbiased estimators under inverse sampling and its extensions. As an application, confidence intervals for the population mean and formulas for determining the number of rare sampled units are proposed. Finally, we introduce measures to quantify the uncertainty around the distribution function when data are collected through a non-probability sample.