Bootstrapping process

The bootstrapping method or bootstrap method (rarely Münchhausen method ) is a method of resampling in statistics . Statistics are repeatedly calculated based on just a sample. Bootstrap methods are used when the theoretical distribution of the statistics of interest is not known. This method was first described by Bradley Efron in 1979 in Bootstrap Methods: Another Look at the Jackknife .

The bootstrap usually replaces the theoretical distribution function of a random variable with the empirical distribution function (relative cumulative frequency function) of the sample . ${\displaystyle F}$ ${\displaystyle {\hat {F}}}$ ${\displaystyle x_{1},\ldots ,x_{n}}$

Action

For this, in the simplest case, bootstrap samples are generated by taking a value with replacement from the given sample for each drawing . This corresponds to the repeated extraction of random numbers from the empirical distribution function . For each bootstrap sample, the value of the statistic of interest is calculated. The distribution of is finally approximated by the empirical distribution of the values . ${\displaystyle B}$ ${\displaystyle x_{b}=(x_{1}^{*},\ldots ,x_{n}^{*})\,,b=1,\ldots ,B}$ ${\displaystyle n}$ ${\displaystyle {\hat {F}}}$ ${\displaystyle T_{b}(x_{1}^{*},\ldots ,x_{n}^{*})=T(x_{b})}$ ${\displaystyle T}$ ${\displaystyle T(X_{1},\ldots ,X_{n})}$ ${\displaystyle B}$ ${\displaystyle T_{b}(x_{1}^{*},\ldots ,x_{n}^{*})}$

In less intuitive models, it is not just repeated pulling from the data that is already available. In terms of method, the bootstrap method can also proceed in such a way that certain parameters of the unknown distribution are estimated and data is regenerated on the basis of this information by generating a distribution with the estimated values.

Web links

• Edition of the journal Statistical Science on the occasion of the 25th anniversary of the bootstrap method

literature

• Bradley Efron: Bootstrap Methods: Another Look at the Jackknife . In: The Annals of Statistics . 7, No. 1, 1979, pp. 1-26. doi : 10.1214 / aos / 1176344552 .
• B. Efron, RJ Tibshirani: An introduction to the bootstrap , New York: Chapman & Hall, 1993
• J. Shao, D. Tu: The Jackknife and Bootstrap . Springer, 1995.