An Elementary Introduction to Maximum Likelihood Estimation for Multinomial Models: Birch's Theorem and the Delta Method

Abstract

A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second-year graduate course in categorical data analysis, can be based on Birch's theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch's theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao's (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.

Key Words:

• Multinomial models
• Maximum likelihood
• Birch's theorem
• Delta method
• Asymptotic distributions