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Zbl 1112.62054
Wu, Xiaoyong; Zou, Guohua; Chen, Jianwei
Unbiased invariant minimum norm estimation in generalized growth curve model. (English)
[J] J. Multivariate Anal. 97, No. 8, 1718-1741 (2006). ISSN 0047-259X

Summary: This paper considers the generalized growth curve model $Y= \sum^m_{i=1}X_iB_i Z_i'+U{\cal E}$ subject to $$R(X_m)\subseteq R(X_{m-1})\subseteq\cdots\subseteq R(X_1),$$ where $B_i$ are the matrices of unknown regression coefficients, $X_i, Z_i$ and $U$ are known covariate matrices, $i=1,2,\dots,m$, and ${\cal E}$ splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix $\Sigma$. An unbiased invariant minimum norm quadratic estimator $(MINQE(U,I))$ of $\text{tr}(C \Sigma)$ is derived and the conditions for its optimally under the minimum variance criterion are investigated. The necessary and sufficient conditions for $MINQE(U,I)$ of $\text {tr}(C\Sigma)$ to be a uniformly minimum variance invariant quadratic unbiased estimator $(UMVIQUE)$ are obtained. An unbiased invariant minimum norm quadratic plus linear estimator $(MINQLE(U,I))$ of $\text{tr}(C\Sigma)+\sum^m_{i=1}\text {tr}(D_i'B_i)$ is also given. To compare with the existing maximum likelihood estimator (MLE) of $\text{tr}(C\Sigma)$, we conduct some simulation studies which show that our proposed estimator performs very well.

MSC 2000:: *62H12 Multivariate estimation
62J99 Linear statistical inference
62J05 Linear regression

Keywords: generalized growth curve model; MINQE(U, I); MINQLE(U,I); UMVIQUE

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