Resampling methods: More Bootstrap background
In the case that \( \widehat{\theta} \) has
more than one component, and the components are independent, we use the
same estimator on each component separately. If the probability
density function of \( X_i \), \( p(x) \), had been known, then it would have
been straight forward to do this by:
- Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).
- Then using these numbers, we could compute a replica of \( \widehat{\theta} \) called \( \widehat{\theta}^* \).
By repeated use of (1) and (2), many
estimates of \( \widehat{\theta} \) could have been obtained. The
idea is to use the relative frequency of \( \widehat{\theta}^* \)
(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \).