Bootstrap methods for constructing standard errors and confidence intervals

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• Suppose we want to know the statistical behavior of the estimator $\hat\theta({y_{1:N}})$ for models in a neighborhood of the MLE, $\theta^*=\hat\theta(y_{1:N}^*)$.

• In particular, let’s consider the problem of estimating uncertainty about $\theta_1$. We want to assess the behavior of the maximum likelihood estimator, $\hat\theta({y_{1:N}})$, and possibly the coverage of an associated confidence interval estimator, $\big[\hat\theta_{1,\mathrm lo}({y_{1:N}}),\hat\theta_{1,\mathrm hi}({y_{1:N}})\big]$. The confidence interval estimator could be constructed using either the Fisher information method or the profile likelihood approach.

• The following simulation study lets us address the following goals:
  
  

  1. Evaluate the coverage of a proposed confidence interval estimator, $[\hat\theta_{1,\mathrm lo},\hat\theta_{1,\mathrm hi}]$,
     
     
  2. Construct a standard error for $\theta_1^*$,
     
     
  3. Construct a confidence interval for $\theta_1$ with exact local coverage.

1. Generate $J$ independent Monte Carlo simulations, $$Y_{1:N}^{[j]} \sim f_{Y_{1:N}}(y_{1:N}\, ; \,\theta^*)\mbox{ for } j\in 1:J.$$

2. For each simulation, evaluate the maximum likelihood estimator, $$\theta^{[j]} = \hat\theta\big(Y_{1:N}^{[j]}\big)\mbox{ for } j\in 1:J,$$ and, if desired, the confidence interval estimator, $$\big[\theta^{[j]}_{1,\mathrm lo},\theta^{[j]}_{1,\mathrm hi}\big] = \big[\hat\theta_{1,\mathrm lo}({X^{[j]}_{1:N}}),\hat\theta_{1,\mathrm hi}({X^{[j]}_{1:N}})\big].$$

3. We can use these simulations to obtain solutions to our goals for uncertainty assessment:
   
   

   1. For large $J$, the coverage of the proposed confidence interval estimator is well approximated, for models in a neighborhood of $\theta^*$, by the proportion of the intervals $\big[\theta^{[j]}_{1,\mathrm lo},\theta^{[j]}_{1,\mathrm hi}\big]$ that include $\theta_1^*$.
      
      
   2. The sample standard deviation of $\{ \theta^{[j]}_1, j\in 1:J\}$ is a natural standard error to associate with $.
      
      
   3. For large $J$, one can empirically calibrate a 95% confidence interval for $\theta_1$ with exactly the claimed coverage in a neighborhood of $\theta^*$. For example, using profile methods, one could replace the cutoff 1.92 by a constant $\alpha$ chosen such that 95% of the profile confidence intervals computed for the simulations cover $\theta_1^*$.

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