Definition:Bootstrapping
Definition
Bootstrapping is a method for obtaining information about the distribution of a population by taking a random sample of $n$ observations, then from this random sample forming further random samples from this.
Bootstrap Sample
These secondary random samples are called bootstrap samples.
They will also typically be of $n$ observations.
They are obtained by sampling with replacement.
Motivation
Bootstrapping is useful when either:
- there is insufficient information to specify the distribution of the population
or:
- there is little analytic theory about properties or estimators.
Refinements are available to improve the estimates exemplified here.
In practice, a statistical software package is generally used, with a reliable random number generator.
Examples
Arbitrary Example
Let $B$ bootstrap samples be taken of a population.
Let $m$ be the median of the population as a whole.
Let ${m_b}^*$ be the median of the $b$th bootstrap sample, where $b = 1, 2, \ldots, B$.
Then the bootstrap estimated standard error of $m$ is given by:
- $\map {\operatorname {se} } m = \sqrt {\dfrac 1 {B - 1} \ds \sum_b \paren { {m_b}^* - {\overline m}^*}^2}$
where ${\overline m}^*$ is the mean of the ${m_b}^*$.
Approximating $95 \%$ confidence limits for the population median are given by the $0 \cdotp 025$ and $0 \cdotp 975$ quantiles of the set of $b$ values ${m_b}^*$.
In practice, good values of $\map {\operatorname {se} } m$ can be obtained with $B = 50$, but values of $B = 1000$ or $B = 2000$ are needed for reliable estimates of confidence intervals.
Also see
- Results about bootstrapping can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bootstrap
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bootstrap