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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

Just a note

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