Balanced Bootstraps

Intermediary between the complete enumeration of all samples and the drawing of a few samples at random lies the Balanced Bootstrap, (see Gleason, 1988, American Statistician,263-266) for a simple algorithm.

The idea is that as when we count all $\vert C_n\vert$ samples each observation has the same chance of appearing, we would like to have each observation appearing overall the same number of times after all $B$ bootstrap resamples have been drawn.

We can be ensured of this by taking a list of $B$ repetitions of each of the observations $x_1,x_2,\ldots, x_n$ and repeating them $B$ times, take a permutation of these $nB$ values and then take the first $n$ as the first sample, and so on.