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Unbiased invariant minimum norm estimation in generalized growth curve model. (English) Zbl 1112.62054

Summary: This paper considers the generalized growth curve model \(Y= \sum^m_{i=1}X_iB_i Z_i'+U{\mathcal E}\) 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 \({\mathcal E}\) 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:

62H12 Estimation in multivariate analysis
62J99 Linear inference, regression
62J05 Linear regression; mixed models
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References:

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