Resampling methods: Bootstrap steps

The independent bootstrap works like this:

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

When you are done, you can draw a histogram of the relative frequency of \( \widehat \theta^* \). 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 \( \widehat{\theta}^* \). Instead you use the estimators corresponding to the statistic of interest. For example, if you are interested in estimating the variance of \( \widehat \theta \), apply the etsimator \( \widehat \sigma^2 \) to the values \( \widehat \theta ^* \).

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