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Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model. (English) Zbl 1157.62035

Summary: Consider the generalized growth curve model \(Y=\sum_{i=1}^m X_iB_iZ_i^\prime +U\mathcal E\) subject to \(R(X_m)\subseteq \cdots \subseteq R(X_{1})\), where \(B_i\) are the matrices of unknown regression coefficients, and \(\mathcal E=(\varepsilon_1,\dots ,\varepsilon_s)^\prime \) and \(\varepsilon_j\) \((j=1,\dots,s)\) are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix \(\Sigma \). We derive necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function tr\((C\varSigma)\) with \(C\geq 0\) exists. Necessary and sufficient conditions for a nonnegative quadratic unbiased estimator \(\mathbf y^\prime A\mathbf y\) with \(\mathbf y = \text{Vec}(Y^\prime)\) of tr\((C\Sigma)\) to be the UMVNNQUE are obtained as well.

MSC:

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
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