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  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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