  Citation: Conti, P.L.; Mecatti, F. Resampling under Complex Sampling Designs: Roots, Development and the Way Forward. Stats 2022, 5, 258–269. https:// doi.org/10.3390/stats5010016 Academic Editor: Wei Zhu Received: 27 January 2022 Accepted: 1 March 2022 Published: 8 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Review Resampling under Complex Sampling Designs: Roots, Development and the Way Forward Pier Luigi Conti 1, * and Fulvia Mecatti 2 1 Dipartimento di Scienze Statistiche, Sapienza Università di Roma, P.le Aldo Moro, 5, 00185 Roma, Italy 2 Dipartimento di Sociologia e Ricerca Sociale, Università di Milano-Bicocca, Via Bicocca Degli Arcimboldi, 8, 20126 Milano, Italy; fulvia.mecatti@unimib.it * Correspondence: pierluigi.conti@uniroma1.it Abstract: In the present paper, resampling for finite populations under an iid sampling design is reviewed. Our attention is mainly focused on pseudo-population-based resampling due to its properties. A principled appraisal of the main theoretical foundations and results is given and discussed, together with important computational aspects. Finally, a discussion on open problems and research perspectives is provided. Keywords: resampling; bootstrap; pseudo-population; asymptotics; empirical processes 1. Introduction 1.1. Generalities Resampling methods have a long and honorable history, going back at least to the seminal paper by [1]. Survey data are an ideal context to use resampling methods to approximate the sampling distribution of statistics, due to both (i) a generally large sample size and (ii) data of typically good quality. The present paper does not aim at providing a complete review of resampling methods in sampling statistics; the interested reader is referred, for instance, to [2]. We mainly focus on a special class of resampling methods—namely those based on pseudo-populations. There are several reasons to support this restriction. First of all, they may be viewed, in many respects, as the “natural” extension of classical Efron’s bootstrap to sampling finite populations, in both descriptive and analytic inference (i.e., inference on finite population and superpopulation parameters, respectively). In the second place, in our knowledge, they are the only methods with a rigorous asymptotic justification in terms of weak convergence of empirical processes, allowing results not only for linear estimators but also for non-linear ones (under suitable differentiability conditions). In extreme synthesis, virtually all resampling methodologies used in sampling from finite populations are based on the idea of accounting for the effect of the sampling design. As it will be seen in the sequel, the main effect of the sampling design is that data cannot be generally assumed independent and identically distributed (i.i.d.). A large portion of the literature on resampling from finite populations focuses on estimating the variance of estimators. The main approaches are essentially the ad hoc approach and plug in approach. The basic idea of the ad hoc approach consists in maintaining Efron’s bootstrap as a resampling procedure but in properly rescaling data in order to account for the dependence among units. This approach is used, among others, in [3,4], where the re-sampled data produced by the “usual” i.i.d. bootstrap are properly rescaled, as well as in [5,6]; cfr. also the review in [7]. In [8] a “rescaled bootstrap process” based on asymptotic arguments is proposed. Among the ad hoc approaches, we also classify [9] (based on a rescaling of weights) and the “direct bootstrap” by [10]. Stats 2022, 5, 258–269. https://doi.org/10.3390/stats5010016 https://www.mdpi.com/journal/stats