Some results on maximum likelihood estimation under the EM algorithm: Asymptotic properties and consistent sandwich estimator of covariance matrix

Although it has been well accepted that the asymptotic covariance matrix of maximum likelihood estimates (MLE) for complete data is given by the inverse Fisher information, this paper shows that when the MLE for incomplete data is derived using the EM algorithm, the asymptotic covariance matrix is however no longer specified by the inverse Fisher information. In general, the new information is smaller than the latter in the sense of Loewner partial ordering. A sandwich estimator of covariance matrix is developed based on the observed information of incomplete data and a consistent estimator of complete-data information matrix. The observed information simplifies calculation of conditional expectation of outer product of the complete-data score function appeared in the Louis (1982) general matrix formula. The proposed sandwich estimator takes a different form than the Huber sandwich estimator under model misspecification framework (Freedman, 2006 and Little and Rubin, 2020). Moreover, it does not involve the inverse observed Fisher information of incomplete data which therefore notably gives an appealing feature for application. Recursive algorithms for the MLE and the sandwich estimator of covariance matrix are presented. Application to parameter estimation of regime switching conditional Markov jump process is considered to verify the results. The simulation study confirms that the MLEs are accurate and consistent having asymptotic normality. The sandwich estimator produces standard errors of the MLE which are closer to their analytic values than those provided by the inverse observed Fisher information.