Bootstrapping populations

Starting with a sample

 , a parametric inference problem consists of computing suitable values – call them estimates – of these parameters precisely on the basis of the sample. An estimate is suitable if replacing it with the unknown parameter does not cause major damage in next computations. In Algorithmic inference, suitability of an estimate reads in terms of compatibility with the observed sample.

In this framework, resampling methods are aimed at generating a set of candidate values to replace the unknown parameters that we read as compatible replicas of them. They represent a population of specifications of a random vector

The rationale of the algorithms computing the replicas, which we denote population bootstrap procedures, is to identify a set of statistics

 , by definition. The 

 , in turn. Then, by plugging the second expression in the former, we obtain