Bootstrapping/Examples

Examples of Bootstrapping

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.