The bootstrap, like the jackknife, is a resampling of the N data points xi
. Whereas jackknife considers N new data
sets, each of containing all the original data points minus 1, bootstrap uses Nboot data sets each containing N points
obtained by random (Monte Carlo) sampling of the original set of N points. During the Monte Carlo sampling, the
probability that a data point is picked is 1/N irrespective of whether it has been picked before. (In the statistics
literature this is called picking from a set “with replacement”.) Hence a given data point xi will, on average, appear
once in each Monte Carlo-generated data set, but may appear not at all, or twice, and so on. The probability that xi
appears ni times is close to a Poisson distribution with mean unity. However, it is not exactly Poissonian because of
the constraint in Eq. (34) below. It turns out that we shall need to include the deviation from Poisson distribution
even for large N. We shall use the term “bootstrap” data sets to denote the Monte Carlo-generated data sets.
More precisely, let us suppose that the number of times xi appears in a Monte Carlo-generated data set is ni
. Since
each bootstrap dataset contains exactly N data points, we have the constraint