## 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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