A note on the existence of the maximum likelihood estimate in variance components models

TL;DR

This paper examines the conditions for the existence of maximum likelihood and REML estimates in variance components models, correcting previous proofs and providing new insights into their theoretical foundations.

Contribution

It identifies errors in prior proofs regarding estimate existence and offers corrected and new proofs for key theorems in variance components models.

Findings

Corrected proof of Theorem 3.1 on MLE existence

New proof of Theorem 3.4 on REML estimate existence

Clarified conditions for estimate existence in variance components models

Abstract

In the paper, the problem of the existence of the maximum likelihood estimate and the REML estimate in the variance components model is considered. Errors in the proof of Theorem 3.1 in the article of Demidenko and Massam (Sankhy\=a 61, 1999), giving a necessary and sufficient condition for the existence of the maximum likelihood estimate in this model, are pointed out and corrected. A new proof of Theorem 3.4 in the Demidenko and Massam's article, concerning the existence of the REML estimate of variance components, is presented.