Maximum Likelihood Imputation

TL;DR

This paper extends the h-likelihood framework to enable joint maximum likelihood estimation of all parameters, including variance components, and introduces a one-shot ML imputation method that bypasses the EM algorithm's expectation step.

Contribution

It demonstrates how to form the h-likelihood for joint maximization of all parameters and introduces a single ML imputation approach for missing data that simplifies traditional methods.

Findings

Enables joint ML estimation of fixed and random parameters.

Allows one-shot ML imputation bypassing EM algorithm.

Clarifies differences in predictions for random effects and missing data.

Abstract

Maximum likelihood (ML) estimation is widely used in statistics. The h-likelihood has been proposed as an extension of Fisher's likelihood to statistical models including unobserved latent variables of recent interest. Its advantage is that the joint maximization gives ML estimators (MLEs) of both fixed and random parameters with their standard error estimates. However, the current h-likelihood approach does not allow MLEs of variance components as Henderson's joint likelihood does not in linear mixed models. In this paper, we show how to form the h-likelihood in order to facilitate joint maximization for MLEs of whole parameters. We also show the role of the Jacobian term which allows MLEs in the presence of unobserved latent variables. To obtain MLEs for fixed parameters, intractable integration is not necessary. As an illustration, we show one-shot ML imputation for missing data by treating them as realized but unobserved random parameters. We show that the h-likelihood bypasses the expectation step in the expectation-maximization (EM) algorithm and allows single ML imputation instead of multiple imputations. We also discuss the difference in predictions in random effects and missing data.