Theory of Inference Based on the Likelihood Function
Roderick J. A. Little, Donald B. Rubin · Wiley series in probability and statistics · 2002
Many methods of estimation for incomplete data can be based on the likelihood function under specific modeling assumptions. This chapter reviews basic theory of inference based on the likelihood function and describes how it is implemented in the incomplete-data setting. It begins by considering maximum likelihood and Bayes’ estimation for complete data sets. Only basic results are given. The chapter outlines some basic large-sample properties of maximum likelihood (ML) and Bayes inference. The likelihood for the parameters based on the incomplete data is derived, ML estimates are found by solving the likelihood equation, and the posterior distribution is obtained by incorporating a prior distribution and performing the necessary integrations. Missing values are a form of data coarsening. Heitjan and Rubin (1991) develop a more general theory for coarsened data that includes heaped, censored, and grouped data as well as missing data.