Maximum likelihood method

Abstract

The most popular estimation approach is the maximum likelihood (ML) method. In this chapter, the ML estimator is defined first, and important asymptotic properties of the ML estimator are formulated in Sect. 4.2. Trans- formations of estimators, not only ML estimators, are discussed in Sect. 4.3. To illustrate the ML approach, we consider the ML method in the linear exponential family (Sect. 4.4) and in univariate GLM (Sect. 4.5). A crucial assumption of ML estimation is the correct specification of the underlying statistical model. Therefore, we discuss the consequences of using the ML method in misspecified models in Sect. 4.6. Even if the model is misspecified, it is based on a likelihood, and the resulting estimator is therefore called a quasi maximum likelihood (QML) estimator (for an in-depth discussion, see White, 1982, 1994). The reader should note that QML estimation is different from quasi likelihood (QL) estimation. The latter approach is a generalization of the generalized linear model (McCullagh and Nelder, 1989; Wedderburn, 1974) and requires the correct specification of the first two moments.