## Bootstrapping in the case of complex sampling designs

We already saw many applications where all these samples were drawn completely at random. However, this is often not the case when one has little information on a finite population, or when the data collection is based on a complex survey design. Of course such information is used to draw a sample in such a manner that costs are minimal. In other words, as an example from business statistics: a lot of small- and medium-sized companies exist in Austria but not many large ones. For precise estimates we need all the largest companies (selection probability 1), but the probability of selection of small companies can be much lower. A complex survey design allows us to draw a good sample with minimal costs.

In complex survey sampling, individuals are therefore sampled with known inclusion probabilities from a population of size N to end up with a sample of size n. The inclusion probabilities can differ between strata (partitions of the population...