Teaching about Approximate Confidence Regions Based on Maximum Likelihood Estimation

Abstract

Maximum likelihood (ML) provides a powerful and extremely general method for making inferences over a wide range of data/model combinations. The likelihood function and likelihood ratios have clear intuitive meanings that make it easy for students to grasp the important concepts. Modern computing technology has made it possible to use these methods over a wide range of practical applications. However, many mathematical statistics textbooks, particularly those at the Senior/Masters level, do not give this important topic coverage commensurate with its place in the world of modern applications. Similarly, in nonlinear estimation problems, standard practice (as reflected by procedures available in the popular commercial statistical packages) has been slow to recognize the advantages of likelihood-based confidence regions/intervals over the commonly use “normal-theory” regions/intervals based on the asymptotic distribution of the “Wald statistic.” In this note we outline our approach for presenting, to students, confidence regions/intervals based on ML estimation.

Key Words:

• Asymptotic approximation
• Confidence interval
• Large sample approximation
• Profile likelihood