Some results on maximum likelihood from incomplete data: finite sample properties and improved M-estimator for resampling

TL;DR

This paper investigates the finite sample properties of maximum likelihood estimation from incomplete data, introduces a robust M-estimator with a sandwich covariance estimator, and applies these methods to a regime switching Markov process, demonstrating improved accuracy and stability.

Contribution

It provides explicit formulas for observed Fisher information, develops a recursive inverse matrix algorithm, and introduces a robust M-estimator with a finite-sample covariance estimator for incomplete data.

Findings

Observed information matrices are positive definite and ordered as expected information.

The recursive inverse matrix algorithm improves computational stability.

The M-estimator yields smaller standard errors than ML estimates in finite samples.

Abstract

This paper presents some results on the maximum likelihood (ML) estimation from incomplete data. Finite sample properties of conditional observed information matrices are established. They possess positive definiteness and the same Loewner partial ordering as the expected information matrices do. An explicit form of the observed Fisher information (OFI) is derived for the calculation of standard errors of the ML estimates. It simplifies Louis (1982) general formula for the OFI matrix. To prevent from getting an incorrect inverse of the OFI matrix, which may be attributed by the lack of sparsity and large size of the matrix, a monotone convergent recursive equation for the inverse matrix is developed which in turn generalizes the algorithm of Hero and Fessler (1994) for the Cram\'er-Rao lower bound. To improve the estimation, in particular when applying repeated sampling to incomplete data, a robust M-estimator is introduced. A closed form sandwich estimator of covariance matrix is proposed to provide the standard errors of the M-estimator. By the resulting loss of information presented in finite-sample incomplete data, the sandwich estimator produces smaller standard errors for the M-estimator than the ML estimates. In the case of complete information or absence of re-sampling, the M-estimator coincides with the ML estimates. Application to parameter estimation of a regime switching conditional Markov jump process is discussed to verify the results. The simulation study confirms the accuracy and asymptotic properties of the M-estimator.