13.4 Nonnormal confidence intervals
Section 13.3 created confidence intervals based on two assumptions. The first was that the normal distribution is a reasonable approximation of the true distribution of the maximum likelihood estimator. We know this assumption is asymptotically true but may not hold for small or even moderate samples. Second, it was assumed that when there is more than one parameter, separate confidence intervals should be constructed for each parameter. Separate intervals can be used in cases like the lognormal distribution where the parameter estimates are independent, but in most cases that is not true. When there is high correlation, it is better to postulate a confidence region, which could be done using the asymptotic covariances and a multivariate normal distribution. However, there is an easier method that does not require a normal distribution assumption (though is still based on asymptotic results).
One way to motivate a confidence region is to consider the meaning of the likelihood function. The parameter value that maximizes this function is our best choice. It is reasonable that values of the parameter which produce likelihood function values close to the maximum are good alternative choices for the true parameter value. Thus, for some choice of c, a confidence region for the parameter might be
the set of all parameters for which the loglikelihood ...