Bootstrap methods for constructing standard errors and confidence intervals
Produced in R version 4.4.0.
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:
- Evaluate the coverage of a proposed confidence interval estimator, \([\hat\theta_{1,\mathrm lo},\hat\theta_{1,\mathrm hi}]\),
- Construct a standard error for \(\theta_1^*\),
- Construct a confidence interval for \(\theta_1\) with exact local coverage.
- Evaluate the coverage of a proposed confidence interval estimator, \([\hat\theta_{1,\mathrm lo},\hat\theta_{1,\mathrm hi}]\),
Generate \(J\) independent Monte Carlo simulations, \[Y_{1:N}^{[j]} \sim f_{Y_{1:N}}(y_{1:N}\, ; \,\theta^*)\mbox{ for } j\in 1:J.\]
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, \[ \big[\theta^{[j]}_{1,\mathrm lo},\theta^{[j]}_{1,\mathrm hi}\big] = \big[\hat\theta_{1,\mathrm lo}({X^{[j]}_{1:N}}),\hat\theta_{1,\mathrm hi}({X^{[j]}_{1:N}})\big].\]
We can use these simulations to obtain solutions to our goals for uncertainty assessment:
- 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 \(\theta_1^*\).
- The sample standard deviation of \(\{ \theta^{[j]}_1, j\in 1:J\}\) is a natural standard error to associate with $.
- 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 \(\theta_1^*\).
- 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 \(\theta_1^*\).
