The linear exponential family

Abstract

Several estimating equations belonging to the class of generalized estimating equations for the mean structure, termed GEE1, can be derived as special cases of the pseudo maximum likelihood 1 (PML1) method. PML1 estima- tion is based on the linear exponential family, and this class of distributions is therefore discussed in this chapter. In Sect. 1.1, the linear exponential family is defined in the canonical form with a natural parameter. Moments of the exponential family can be easily obtained by differentiation (Sect. 1.2), which can in turn be used for parameterizing the exponential family in the mean structure (Sect. 1.3). Some properties of the linear exponential family are re- quired for PML1 estimation in Chapt. 5, and they are presented in Sect. 1.4. In Sects. 1.5 and 1.6, several examples for univariate and multivariate distri- butions belonging to the linear exponential family are given to illustrate the broad applicability of the linear exponential family. Finally, the relationship to the parameterization in generalized linear models is established in Sect. 1.7.