Estimability of variance components when all model matrices commute

Rosemary Anne Bailey, Sandra S. Ferreira, Dario Ferreira, Celia Nunes

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


This paper deals with estimability of variance components in mixed models when all model matrices commute. In this situation, it is well known that the best linear unbiased estimators of fixed effects are the ordinary least squares estimators. If, in addition, the family of possible variance-covariance matrices forms an orthogonal block structure, then there are the same number of variance components as strata, and the variance components are all estimable if and only if there are non-zero residual degrees of freedom in each stratum.

We investigate the case where the family of possible variance-covariance matrices, while still commutative, no longer forms an orthogonal block structure. Now the variance components may or may not all be estimable, but there is no clear link with residual degrees of freedom. Whether or not they are all estimable, there may or may not be uniformly best unbiased quadratic estimators of those that are estimable. Examples are given to demonstrate all four possibilities.
Original languageEnglish
Pages (from-to)144-160
Number of pages17
JournalLinear Algebra and its Applications
Early online date17 Dec 2015
Publication statusPublished - 1 Mar 2016


  • Analysis of variance
  • Commutativity
  • Mixed model
  • Orthogonal block structure
  • Segregation


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