CAN Estimators in Exponential and Cramér Families

Chapters 2 and 3 present the theory related to consistent and CAN estimators. Chapter 4 is concerned with the study of a CAN estimator of a parameter, when a probability distribution of X belongs to a specific family of distributions such as an exponential family or a Cramér family. An exponential family is a subclass of a Cramér family. In Sect. 4.2, we prove that in a one-parameter exponential family and in a multiparameter exponential family, the maximum likelihood estimator and the moment estimator based on a sufficient statistic are the same and these are CAN estimators. Section 4.3 presents the Cramér-Huzurbazar theory for the distributions belonging to a Cramér family. Cramér-Huzurbazar theory, which is usually referred to as standard large sample theory of maximum likelihood estimation, asserts that for large sample size, with high probability, the maximum likelihood estimator of a parameter is a CAN estimator. These results are heavily used in Chaps. 5 and 6 to find the asymptotic null distribution of the likelihood ratio test statistic, Wald’s test statistic and the score test statistic. In many models, the system of likelihood equations cannot be solved explicitly and we need some numerical procedures. In Sect. 4.4, we discuss frequently used numerical procedures to solve the system of likelihood equations, such as Newton-Raphson procedure and method of scoring. The last section illustrates the results established in Sects. 4.2 to 4.4 using $$\texttt {R}$$ R software.