For example, imagine you have a sample of 3 observations (1.1, 2.2, 3,3), from which you wish to obtain
( B = ) 4 bootstrap statistics. For simplicity, let us suppose the statistic of interest
( Θ ) is the median
( = 2.2).
A normal (unbalanced) bootstrap might yield the following result.
- 3.3, 3.3, 3.3 so ^{*}_{1} = 3.3
- 2.2, 2.2, 3.3 so ^{*}_{2} = 2.2
- 1.1, 2.2, 2.2 so ^{*}_{3} = 2.2
- 1.1, 2.2, 2.2 so ^{*}_{4} = 2.2
The relative frequencies of the observations in these resamples are
^{2}/_{12} 1.1s, ^{6}/_{12} 2.2s, and ^{3}/_{12} 3.3s - which is rather different from the relative frequencies in our original sample. The average result, where
B approaches infinity, would be the same as our sample - but for any finite number of resamples our random selection causes their relative frequencies to differ from their 'expected' proportions.
For the equivalent balanced bootstrap we would take ( B = ) 4 copies of our sample whose relative frequencies, by definition, would be ^{4}/_{12} 1.1s, ^{4}/_{12} 2.2s, and ^{4}/_{12} 3.3s - and is identical to the sample we want resampled.
Sampling these 12 observations, without replacement, gives us our balanced bootstrap.
- 1.1, 1.1, 3.3 so ^{*}_{1} = 1.1
- 1.1, 2.2, 3.3 so ^{*}_{2} = 2.2
- 1.1, 2.2, 2.2 so ^{*}_{3} = 2.2
- 1.1, 2.2, 3.3 so ^{*}_{4} = 2.2