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### How many different bootstrap samples are there?

By different samples, the samples must differ as sets, ie there is no difference between the sample  , ie the observations are exchangeable or the statistic of interest is a symmetrical function of the sample: .
Definition:
The sequence of random variables is said to be exchangeable if the distribution of the vector is the same as that of , for any permutation of elements.

Suppose we condition on the sample of distinct observations , there are as many different samples as there are ways of choosing objects out of a set of possible contenders, repetitions being allowed.

At this point it is interesting to introduce a new notation for a bootstrap resample, up to now we have noted a possible reasample, say , because of the exchangeability/symmetry property we can recode this as the vector counting the number of occurrences of each of the observations. in this recoding we have and the set of all bootstrap resamples is the dimensional simplex Here is the argument I used in class to explain how big is. Each component in the vector is considered to be a box, there are boxes to contain balls in all, we want to contain to count the number of ways of separating the n balls into the boxes. Put down separators of to make boxes, and balls, there will be positions from which to choose the bars' positions, for instance our vector above corresponds to: oo||o|oo| . Thus Stirling's formula ( ) gives an approximation ,

here is the function file approxcom.m

function out=approxcom(n)
out=round((pi*n)^(-.5)*2^(2*n-1));

that produces the following table of the number of resamples: Are all these samples equally likely, thinking about the probability of drawing the sample of all 's by choosing the index  times in the integer uniform generation should persuade you that this sample appears only once in times. Whereas the sample with once and all the other observations can appear in out of the ways.   Next: Which is the most Up: The combinatorics of the Previous: The combinatorics of the
Susan Holmes 2004-04-27