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# Article

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Keywords:
regression linear model; epoch model; linear epoch regression model; locally best linear unbiased estimators; first order parameters; locally minimum variance quadratic unbiased and invariant estimators; estimable linear function; second order parameters; algorithms; block structure; sparseness of the covariance matrix
Summary:
In the linear epoch regression model $E(Y^{(j,j)}=E \left(\matrix Y_1^{(1)}\\ \vdots\\Y_j^{(j)} \endmatrix\right)=\left(\matrix X_{11}, & X_{21}, & 0, & \dots, & 0\\X_{12}, & 0, & X_{22}, & \dots, & 0\\\vdots & \vdots & \vdots & \ddots & \vdots \\ X_{1j}, & 0, & 0, & \dots & X_{2j} \endmatrix\right)\left(\matrix \beta_1\\ \beta_{21}\\ \vdots\\ \beta_{2j} \endmatrix\right)$, $\text{var}\left(Y^{(j,j)\right)= \left(\matrix \sum^{p1}_{s1}\vartheta_{1_{s1}}H_{1_{s1}}, & \dots , & 0\\ \vdots & \ddots & \vdots\\ 0, & \dots, & \sum^{p_j}_{s_j}\vartheta_{js_j}H_{js_j} \endmatrix \right)$ the locally best linear unbiased estimators of the first order parameters and the locallz minimum variance quadratic unbiased and invariant estimators of an unbiasedly and invariantly estimable linear function of the second order parameters in the $jth$ epoch and after the $jth$ epoch are derived. The algorithms mentioned utilize the special block structure of the model and the sparseness of the covariance matrix of the observation vector.
References:
[1a] Lubomír Kubáček: Foundations of Estimation Theory. Elsevier, Amsterdam, Oxford, New York, Tokyo, 1988. MR 0995671
[2a] Lubomír Kubáček: Special structures of mixed linear model with nuisance parameters. Math. Slovaca 40 (1990), 191-207. MR 1094773

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