# zbMATH — the first resource for mathematics

A new derivation of BLUPs under random-effects model. (English) Zbl 1329.62264
Summary: This paper considers predictions of vectors of parameters under a general linear model $$y= X\beta + \varepsilon$$ with the random coefficients $$\beta$$ satisfying $$\beta = A\alpha + \gamma$$. It utilizes a standard method of solving constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and obtains the best linear unbiased predictor (BLUP) of given vector $$F\alpha + G\gamma + H\varepsilon$$ of the unknown parameters in the model. Some special cases of the BLUPs are also presented. In particular, a general decomposition equality $$y = \text{BLUE}(XA\alpha) + \text{BLUP}(X\gamma) + \text{BLUP}(\varepsilon)$$ is proved under the random-effects model. A further problem on BLUPs of new observations under the random-effects model is also addressed.

##### MSC:
 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models 62J10 Analysis of variance and covariance (ANOVA)
Full Text:
##### References:
 [1] Harville, D, Extension of the Gauss-Markov theorem to include the estimation of random effects, Ann Stat, 4, 384-395, (1976) · Zbl 0323.62043 [2] Henderson, CR, Best linear unbiased estimation and prediction under a selection model, Biometrics, 31, 423-447, (1975) · Zbl 0335.62048 [3] Jiang, J, A derivation of BLUP-best linear unbiased predictor, Stat Prob Lett, 32, 321-324, (1997) · Zbl 0886.62066 [4] Liu, XQ; Rong, JY; Liu, XY, Best linear unbiased prediction for linear combinations in general mixed linear models, J Multivar Anal, 99, 1503-1517, (2008) · Zbl 1144.62047 [5] Marsaglia, G; Styan, GPH, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292, (1974) [6] Penrose, R, A generalized inverse for matrices, Proc Cambridge Phil Soc, 51, 406-413, (1955) · Zbl 0065.24603 [7] Pfeffermann, D, On extensions of the Gauss-Markov theorem to the case of stochastic regression coefficients, J R Stat Soc Ser B, 46, 139-148, (1984) · Zbl 0546.62038 [8] Robinson, GK, That BLUP is a good thing: the estimation of random effects, Stat Sci, 6, 15-32, (1991) · Zbl 0955.62500 [9] Searle, SR, The matrix handling of BLUE and BLUP in the mixed linear model, Linear Algebra Appl, 264, 291-311, (1997) · Zbl 0889.62059 [10] Tian, Y, More on maximal and minimal ranks of Schur complements with applications, Appl Math Comput, 152, 675-692, (2004) · Zbl 1077.15005 [11] Wu, Q, Several results on admissibility of linear estimate of stochastic regresssion coeffieincts and parameters (in Chinese), Acta Math Appl Sinica, 11, 95-106, (1988) · Zbl 0661.62041 [12] Xu, LW; Yu, SH, Admissible prediction in superpopulation models with random regression coefficients under matrix loss function, J Multivar Anal, 103, 68-76, (2012) · Zbl 1229.62009 [13] Zhan, J; Chen, J, Admissibility of linear estimators of regression coefficients under quadratic loss, Acta Math Appl Sinica, 8, 237-244, (1992) · Zbl 0765.62013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.