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Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model. (English) Zbl 1155.62014

Summary: Let \(Y = X\Theta Z' + {\mathcal E}\) be a growth curve model with \(\mathcal E\) distributed with mean 0 and covariance \(I_n\otimes \Sigma\), where \(\Theta,\Sigma\) are unknown matrices of the parameters and \(X,\;Z\) are known matrices. For the estimable parametric transformation of the form \(\gamma = C\Theta D'\) with given \(C\) and \(D\), the two-stage generalized least-squares estimator \[ \widehat\gamma(Y)=C(X'X)^{-1}X'Y\widehat\Sigma^{-1}(Y)Z(Z'\widehat\Sigma^{-1}(Y)Z)^{-1}D' \]
converges in probability to \(\gamma\) as the sample size \(n\) tends to infinity and, further, \(\sqrt{n}[\widehat\gamma(Y)-\gamma]\) converges in distribution to the multivariate normal distribution \[ {\mathcal N}(0, (CR^{-1} C')\otimes (D(Z'\Sigma^{-1} Z)^{-1}D')) \] under the condition that \(\lim_{n\to\infty} X'X/n=R\) for some positive definite matrix \(R\). Moreover, the unbiased and invariant quadratic estimator
\[ \widehat\Sigma(Y)=Y'WY, \quad W\equiv (n-\operatorname{rank} X)^{-1}(I-P_X),\quad P_X=X(X'X)^-X', \]
is also proved to be consistent with the second-order parameter matrix \(\Sigma\).

MSC:

62F12 Asymptotic properties of parametric estimators
62H12 Estimation in multivariate analysis
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