Resampling methods: More Bootstrap background

In the case that $\widehat{\beta}$ 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 straightforward 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 $\widehat{\beta}$ called $\widehat{\beta}^*$.

By repeated use of the above two points, many estimates of $\widehat{\beta}$ can be obtained. The idea is to use the relative frequency of $\widehat{\beta}^*$ (think of a histogram) as an estimate of $p(\boldsymbol{t})$.

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