Likelihood Inference with Missing Data

Daowen Zhang

Likelihood inference plays a central role in the practice of statistics. In the presence of missing data, likelihood inference is often hampered by intractable integration in the evaluation of the likelihood function for the observed data. The EM (expectation-maximization) algorithm (Dempster, Laird and Rubin, 1977) was proposed to obtain the maximum likelihood estimates of the model parameters in a very attractive way: it updates the parameter estimates in two usually much simpler steps: E and M steps and guarantees an increase of the observed likelihood function. In this seminar, I will describe likelihood inference when data are missing, and I will go through the original EM algorithm and present some of its variations: one step EM, ECM and SEM. One step EM uses one iteration of Newton-Raphson in the M step, which is useful in the situation where the M step is not easy to evaluate. ECM replaces the M step by a series of M steps. SEM uses EM to obtain the asymptotic variance of the MLEs. Some examples will be used to illustrate the EM algorithm, and a list of references will be provided.