ABSTRACT
The approximate behaviour of estimators and other statistics when the sample size becomes large is investigated. The notions of asymptotic consistency and normality are introduced, and it is shown that moment estimators are consistent and asymptotically normally distributed using the delta method in combination with the Central Limit Theorem. In particular, it is shown that the maximum likelihood estimator in regular exponential families—since they are moment estimators—are asymptotically well-defined, consistent, and asymptotically normally distributed with the Fisher information as their asymptotic variance. The results are extended to curved exponential families; here extra work is needed since the maximum likelihood method in such families are not moment estimators. The asymptotic chi-square-distribution of the likelihood ratio statistic and quadratic forms of Wald type is also established.
