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:

1. Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).
2. 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 \( p(\hat{t}) \).

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