Bootstrap Resampling for Quantifying Uncertainty

Bootstrap resampling provides two approaches nonparametric and parametric bootstrap samples. In both we begin with a mle or procedure fitted to n independent 

For example, we can build B = 400 parametric bootstrap samples, each by adding noise to the model for predicting y from the last example. We assume the noise, denoted s has a Gaussian distribution with mean zero and variance = YH=i( yi y 9 In, written as a,- A1(0, ri). Then we can write y = a -1-/3/fc 0. 5) -b Si, where a and /3 are the maximum-UkeUhood estimates of the intercept and slope, respectively. As B increases, the parametric bootstrap samples capture the variability of the estimates for a and a -b /3. For example, as with the nonparametric bootstrap, with B = 400, we can extract a 95% confidence interval for the estimates by selecting the 0.025 and 0.975 quantiles of the bootstrap estimates for each z-value. 

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