We have now seen two different ways to calculate approximate confidence intervals (CI) for an unknown parameter. Previously, we calculated CI based on the asymptotic distribution of the MLE (CI-MLE). Here, we showed how to calculate the CI based on the asymptotic distribution of the deviance (CI-D).
We discussed various differences and pros and cons of the two:
CI-MLE is always symmetric about the MLE. CI-D is not.
CI-MLE can include values with zero likelihood (e.g. infeasible values such as negative probabilities, as seen here). CI-D will only include feasible values.
CI-D is typically harder to calculate than CI-MLE.
For reasons we will not go into here, CI-D is typically more accurate than CI-MLE.
CI-D is invariant to re-parametrization; CI-MLE is not. (This is a good thing for CI-D, that we will learn more about in subsequent lectures).
Overall, CI-D is usually preferred to CI-MLE (since the only disadvantage is that it is harder to compute).
DEVIANCES ARE ALWAYS NON-NEGATIVE!