Iterative Likelihood: A Unified Inference Tool

Abstract

We propose a framework for inference based on an “iterative likelihood function,” which provides a unified representation for a number of iterative approaches, including the EM algorithm and the generalized estimating equations (GEEs). The parameters are decoupled to facilitate construction of the inference vehicle, to simplify computation, or to ensure robustness to model misspecification and then recoupled to retain their original interpretations. For simplicity, throughout the paper, we will refer to the log-likelihood as the “likelihood.” We define the global, local, and stationary estimates of an iterative likelihood and, correspondingly, the global, local, and stationary attraction points of the expected iterative likelihood. Asymptotic properties of the global, local, and stationary estimates are derived under certain assumptions. An iterative likelihood is usually constructed such that the true value of the parameter is a point of attraction of the expected log-likelihood. Often, one can only verify that the true value of the parameter is a local or stationary attraction, but not a global attraction. We show that when the true value of the parameter is a global attraction, any global estimate is consistent and asymptotically normal; when the true value is a local or stationary attraction, there exists a local or stationary estimate that is consistent and asymptotically normal, with a probability tending to 1. The behavior of the estimates under a misspecified model is also discussed. Our methodology is illustrated with three examples: (i) estimation of the treatment group difference in the level of censored HIV RNA viral load from an AIDS clinical trial; (ii) analysis of the relationship between forced expiratory volume and height in girls from a longitudinal pulmonary function study; and (iii) investigation of the impact of smoking on lung cancer in the presence of DNA adducts. Two additional examples are in the supplementary materials, GEEs with missing covariates and an unweighted estimator for big data with subsampling. Supplementary files for this article are available online.

Keywords:

• Asymptotic properties
• Attraction
• Censoring
• EM algorithm
• Generalized estimating equations
• Mean score method

Supplementary Materials

The supplementary material includes the detailed proofs for Theorem 1 in Section 3.2, Theorem 2 in Section 4.1, and additional two examples.

Acknowledgments

The authors thank the Editor, an associate editor and two referees for their insightful suggestions and comments that have greatly improved an earlier version of this paper.

Additional information

Funding

Liang’s research was partially supported by NSF grant DMS-1620898.