13.3 Variance and interval estimation
In general, it is not easy to determine the variance of complicated estimators such as the mle. However, it is possible to approximate the variance. The key is a theorem that can be found in most mathematical statistics books. The particular version stated here and its multiparameter generalization is taken from  and stated without proof. Recall that L(θ) is the likelihood function and l(θ) its logarithm. All of the results assume that the population has a distribution that is a member of the chosen parametric family.
Theorem 13.5 Assume that the pdf (pf in the discrete case) f (x; θ) satisfies the following for θ in an interval containing the true value (replace integrals by sums for discrete variables):