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Simultaneous optimal estimation in linear mixed models

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Abstract

Simultaneous optimal estimation in linear mixed models is considered. A necessary and sufficient condition is presented for the least squares estimator of the fixed effects and the analysis of variance estimator of the variance components to be of uniformly minimum variance simultaneously in a general variance components model. That is, the matrix obtained by orthogonally projecting the covariance matrix onto the orthogonal complement space of the column space of the design matrix is symmetric, each eigenvalue of the matrix is a linear combinations of the variance components and the number of all distinct eigenvalues of the matrix is equal to the the number of the variance components. Under this condition, uniformly optimal unbiased tests and uniformly most accurate unbiased confidence intervals are constructed for the parameters of interest. A necessary and sufficient condition is also given for the equivalence of several common estimators of variance components. Two examples of their application are given.

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Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, 100124, Beijing, China

    Mi-Xia Wu

  2. Division of Epidemiology, Statistics and Prevention Research, Eunice Kennedy Shriver, National Institute of Child Health and Human Development, NIH/DHHS 6100 Executive Boulevard, Rockville, MD, 20852, USA

    Kai-Fun Yu & Aiyi Liu

  3. School of Statistics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, China

    Tie-Feng Ma

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  1. Mi-Xia Wu
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  2. Kai-Fun Yu
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  3. Aiyi Liu
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  4. Tie-Feng Ma
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Correspondence to Mi-Xia Wu.

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Wu, MX., Yu, KF., Liu, A. et al. Simultaneous optimal estimation in linear mixed models. Metrika 75, 471–489 (2012). https://doi.org/10.1007/s00184-010-0337-1

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  • DOI: https://doi.org/10.1007/s00184-010-0337-1

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