Some results on maximum likelihood of incomplete data: finite sample properties, consistent sandwich estimator of covariance matrix and recursive algorithms

This paper presents some new results on maximum likelihood of incomplete data. Finite sample properties of conditional observed information matrices are established. In particular, they possess the same Loewner partial ordering properties as the expected information matrices do. In its new form, the observed Fisher information (OFI) simplifies conditional expectation of outer product of the complete-data score function appearing in the Louis (1982) general matrix formula. It verifies positive definiteness and consistency to the expected Fisher information as the sample size increases. Furthermore, it shows a resulting information loss presented in the incomplete data. For this reason, the OFI may not be the right (consistent and efficient) estimator to derive the standard error (SE) of maximum likelihood estimates (MLE) for incomplete data. A sandwich estimator of covariance matrix is developed to provide consistent and efficient estimates of SE. The proposed sandwich estimator coincides with the Huber sandwich estimator for model misspecification under complete data (Huber, 1967; Freedman, 2006; Little and Rubin, 2020). However, in contrast to the latter, the new estimator does not involve OFI which notably gives an appealing feature for application. Recursive algorithms for the MLE, the observed information and the sandwich estimator are presented. Application to parameter estimation of a regime switching conditional Markov jump process is considered to verify the results. The recursive equations for the inverse OFI generalizes the algorithm of Hero and Fessler (1994). The simulation study confirms that the MLEs are accurate and consistent having asymptotic normality. The sandwich estimator produces standard error of the MLE close to their analytic values compared to those overestimated by the OFI.