Resampling methods: Bootstrap steps

The independent bootstrap works like this:

1. Draw with replacement $n$ numbers for the observed variables $\boldsymbol{x} = (x_1,x_2,\cdots,x_n)$.
2. Define a vector $\boldsymbol{x}^*$ containing the values which were drawn from $\boldsymbol{x}$.
3. Using the vector $\boldsymbol{x}^*$ compute $\widehat{\beta}^*$ by evaluating $\widehat \beta$ under the observations $\boldsymbol{x}^*$.
4. Repeat this process $k$ times.

When you are done, you can draw a histogram of the relative frequency of $\widehat \beta^*$. This is your estimate of the probability distribution $p(t)$. Using this probability distribution you can estimate any statistics thereof. In principle you never draw the histogram of the relative frequency of $\widehat{\beta}^*$. Instead you use the estimators corresponding to the statistic of interest. For example, if you are interested in estimating the variance of $\widehat \beta$, apply the etsimator $\widehat \sigma^2$ to the values $\widehat \beta^*$.

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