Use of the pseudo-population approach, despite its many theoretical merits, is held back by its computational complexity. Real populations could contain millions of people, and thus the 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 points are 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∗is is in [27], where non-integer N∗is are allowed via the Horvitz–Thompson-based bootstrap (HTB) method. However, unless the sampling fraction n/N tends to 0 as N and n increase, HTB does not generally possess the good asymptotic properties outlined in the previous sections.

An interesting computational shortcut is in [28], 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], but based on correct approximations of first order inclusion probabilities, were developed in [29] 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.

The pseudo-population approach, despite its merits, requires further development from both the theoretical and computational 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 development as well as the consideration of non-respondent units.

Again, from a theoretical perspective, a major issue is the development of theoretically sound resampling methodologies for informative sampling designs. The major drawback is that, apart from the exception of adaptive designs (cfr. [30]) 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] and the references therein).

From a computational point of view, as indicated earlier, the computational shortcuts developed thus far only work in 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.