2. Accounting for the Sampling Design in Resampling: The Pseudo-Population Approach

Among several techniques that aim at accounting for the sampling design in resampling from finite populations, we consider here the approach based on pseudo-populations. The idea of pseudo-population goes back, at least, to ^[11] in the case of median estimation essentially under srs when the population size is a multiple of the sample size. Rather similar ideas are in ^[12] for srs, again under the condition that the ratio between population size and sample size is a ninteger, and in ^[13], for stratified random sampling. A major step forward is the paper by ^[14], where the construction of a pseudo-population is studied under a general πps sampling design, with general first order inclusion probabilities. In [19]^[15], a different approach to the construction of a pseudo-population, very interesting in many respects, is considered. The pseudo-population approach to resampling can be considered as a two-phase procedure. In the first phase, a pseudo-population (roughly speaking, a prediction of the population) is constructed. In the second phase, a (bootstrap) sample is drawn from the pseudo-population. Broadly speaking, this approach parallels the plug-in principle by Efron. The pseudo-population is plugged in the sampling process and is used as a “surrogate” of the actual finite population. In the second phase, a sample is drawn from the pseudo-population, according to a sampling design that mimics the original one. In this view, the pseudo-population mimics the real population, and the (re)sampling process from the pseudo-population mimics the (original) sampling process from the real population.

2.1. Resampling from Pseudo-Populations

ResampliIng based^[16], onit pseudo-populations actually parallels Efron’s bootstrap for i.i.d. is thoroughly illustrated a class of resampling techniques for finite pobservpulations. The basic ideas are relatively simple, once the problem is approached in terms of an appropriate estimator under a general (complex) sampling design which is asymptotically ofcorrect thunde f.p.d.f.

2.2. Resampling Based on Pseudo-Populations: Basics Results for Descriptive Inference

Ther mild assumain theoreptions. Practical justification for resampling based on recommendations for finite sample sizes and on how to construct the pseudo-population is of asymptotic nature, similar, in many respects, to results in [17] for Efron’s bootstrapare also given.

3. Computational Issues

UseApplication of the pseudo-population approach, despite its many theoretical merits, is held backcan be limited in practice by its computational complexityburden. Real populations could contain millions of peopleunits, and thus the cactual construction of a pseudo-population could be computationally cumbersome. For this reason, it is of primary interest to develop shortcuts that, while possessing the fundamental theoretical properties described in the above sections, are computationally simple to implement because they avoid the physical construction of the pseudo-population. The above poiInts are^[17] thoroughly discussed in [26], where the problem of resampling for finite populations is addressed as a problem of sampling with replacement directly from the sample data, the original sample, henceforth, with different drawing probabilities. An attempt to avoid complications related to integer-valued N∗i bootstrap weights, i.e. the number of replications of each sampled is iunit into the pseudo-population, is offered in [27]^[18], where non-integer N∗i bootstrap weights are allowed via  via the Horvitz–Thompson-based bootstrap (HTB) method. However, unless the sampling fraction n/N  tends to 0 as both population s N ize and n  sample size increase, HTB does not generally possess the good asymptotic properties outlined in the^[16]. previous sections. An interesting computational shortcut is in [28]^[19], where the pseudo-population (again with possibly non-integer N∗is) is only implicitly used, and a computational scheme based on drawings with replacements from the original sample is proposed. Unfortunately, although the main idea behind that paper is interesting, the proposed bootstrap method fails to possess good asymptotic properties. Computational shortcuts, based on ideas similar to those in [28]^[19], but based on correct approximations of first order inclusion probabilities, were developed in [29]^[20] for descriptive, design-based inference. In particular, in that paper, methodologies based on drawings with replacements from the original sample were proposed, and their merits, from both a theoretical and a computational point of view, were studied. As remarked by a referee, another drawback of the pseudo-population approach is the apparent necessity to generate and save a large number of bootstrap sample files. However, it is not necessary to save all the bootstrap sample files. Only the original sample file must be saved along with two additional variables for each bootstrap replicate: one variable that contains the number of times each sample unit is used to create the pseudo-population and another one containing the number of times each sample unit has been selected in the bootstrap sample. In other words, it can be implemented similar to methods that rescale the sampling weights.

Another practical drawback related to the pseudo-population approach is the seeming necessity to generate and store a large number of bootstrap sample files. However, it is not necessary to save all the bootstrap sample files. Only the original sample file should be saved along with two additional variables for each bootstrap replicate: one variable that contains the number of times each sample unit is used to create the pseudo-population and another one containing the number of times each sample unit has been selected in the bootstrap sample. In other words, it can be implemented similarly to methods that rescale the sampling weights.

4. Open Problems and Final Considerations

The pseudo-population approach, despite its merits, requires further development from both the theoretical and computational perspectives.

The pseudo-population app Froach, despite its merits, requires further development from both the theoretical and computational p a theoretical perspectives. From a theoretical point of view, the results obtained thus far only refer to non-informative single-stage designs. The consideration of multi-stage designs appears as a necessary dedevelopment as well as the consideration of non-respondent units. Again, from a t of theoretical perspective, a major issue is the development of theoretically sound resampling methodologies for informative sampling designs is a major issue calling for more research. The majorin drawback is that, apart from the exception of adaptive designs (cfr. [30]^[21]) and the references therein) first order inclusion probabilities can rarely be computed, as these might depend on unobserved quantities. This is what happens, for instance, with most of the network sampling designs that are actually used for hidden populations, where the inclusion probabilities are unknown and depend on unobserved/unknown network links (cfr. [30,31]^[21][22] and the references therein). F Again from athe computationtheoretical point of view, as indicated earlier, the computational shortcuts developed thus far only work in the case of descriptive inference. The the consideration of multistage designs appears as a further necessary development of theoretically well-founded computational schemes valid for analytic inference is an important issue that deserves further attentionas well as the consideration of non-respondent units.

From a computational point of view, the computational shortcuts developed thus far only apply to the case of descriptive inference. The development of theoretically well-founded  computational schemes valid for analytic inference is an important issue that deserves further attention.