Abstract
In this paper we consider two mixed linear models, \({\fancyscript{M}_1}\) and \({\fancyscript{M}_2}\), say, which have different covariance matrices. We review some useful concepts and results on the best linear unbiased estimators (BLUEs) and on best linear unbiased predictors (BLUPs). We give new necessary and sufficient conditions, without making any rank assumptions, that every representation of the BLUP of the random effect under the model \({\fancyscript{M}_1}\) continues to be BLUP under the model \({\fancyscript{M}_2}\). These considerations are generalized to two linear models with new unobserved future observations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baksalary JK (2004) An elementary development of the equation characterizing best linear unbiased estimators. Linear Algebra Appl 388: 3–6
Christensen R (2002) Plane answers to complex questions: the theory of linear models. 3. Springer, New York
Goldberger AS (1962) Best linear unbiased prediction in the generalized linear regression model. J Am Stat Assoc 58: 369–375
Haslett J, Haslett SJ (2007) The three basic types of residuals for a linear model. Int Stat Rev 75: 1–24
Haslett SJ, Puntanen S (2010) Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat Pap 51. doi:10.1007/s00362-009-0219-7
Henderson CR (1950) Estimation of genetic parameters. Ann Math Stat 21: 309–310
Henderson CR (1963) Selection index and expected genetic advance. Statistical Genetics and Plant Breeding, National Research Council Publication No. 982. National Academy of Sciences, Washington, pp 141–163
Isotalo J, Puntanen S (2006) Linear prediction sufficiency for new observations in the general Gauss–Markov model. Commun Stat Theory Methods 35: 1011–1023
Isotalo J, Möls M, Puntanen S (2006) Invariance of the BLUE under the linear fixed and mixed effects models. Acta et Commentationes Universitatis Tartuensis de Mathematica 10: 69–76
Isotalo J, Puntanen S, Styan GPH (2008) A useful matrix decomposition and its statistical applications in linear regression. Commun Stat Theory Methods 37: 1436–1457
Kala R (1981) Projectors and linear estimation in general linear models. Commun Stat Theory Methods 10: 849–873
Mitra SK, Moore BJ (1973) Gauss–Markov estimation with an incorrect dispersion matrix. Sankhyā, Ser A 35: 139–152
Möls M (2004) Linear mixed models with equivalent predictors. PhD Thesis. Dissertationes Mathematicae Universitatis Tartuensis, 36
Puntanen S, Styan GPH (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator (with discussion). The American Statistician 43:151–161 (Commented by O. Kempthorne on pp. 161–162 and by S. R. Searle on pp. 162–163, Reply by the authors on p. 164)
Rao CR (1967) Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Le Cam LM, Neyman J (eds) Proceedings of the fifth Berkeley symposium on mathematical statistics and probability: Berkeley, California, 1965/1966, vol. 1, University of California Press, Berkeley, pp 355–372
Rao CR (1968) A note on a previous lemma in the theory of least squares and some further results. Sankhyā, Ser A 30: 245–252
Rao CR (1971) Unified theory of linear estimation. Sankhyā, Ser A 33:371–394. (Corrigendum (1972), 34, p. 194 and p. 477)
Rao CR (1973) Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J Multivar Anal 3: 276–292
Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York
Rao CR, Toutenburg H, Shalabh, Heumann C (2008) Linear models and generalizations: least squares and alternatives. 3. Springer, New York
Robinson GK (1991) That BLUP is a good thing: the estimation of random effects (with discussion on pp. 32–51). Stat Sci 6: 15–51
Roy A (2008) Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up. J Appl Stat 35: 307–320
Searle SR (1997) The matrix handling of BLUE and BLUP in the mixed linear model. Linear Algebra Appl 264: 291–311
Zyskind G (1967) On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. Ann Math Stat 38: 1092–1109
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Haslett, S.J., Puntanen, S. On the equality of the BLUPs under two linear mixed models. Metrika 74, 381–395 (2011). https://doi.org/10.1007/s00184-010-0308-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-010-0308-6