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replicated regression model; best unbiased estimators
The aim of the paper is to estimate a function $\gamma=tr(D\beta\beta')+tr(C\sum)$ (with $d, C$ known matrices) in a regression model $(Y, X\beta,\sum)$ with an unknown parameter $\beta$ and covariance matrix $\sum$. Stochastically independent replications $Y_1,\ldots, Y_m$ of the stochastic vector $Y$ are considered, where the estimators of $X\beta$ and $\sum$ are $\bar{Y}=\frac 1 m \sum ^m _{i=1} Y_i$ and $\hat{\sum}=(m-1)^{-1} \sum^m_{i=1}(Y_i-\bar{Y})(Y_i-\bar{Y})'$, respectively. Locally and uniformly best inbiased estimators of the function $\gamma$, based on $\bar{Y}$ and $\hat{\sum}$, are given.
[1] Jürgen Kleffe: C. R. Rao's MINQUE for replicated and multivariate observations. Lecture Notes in Statistics 2. Mathematical Statistics and Probability Theory. Proceedings Sixth International Conference. Wisla (Poland) 1978. Springer N. York, Heidelberg, Berlin 1979, 188-200.
[2] Jürgen Kleffe, Júlia Volaufová: Optimality of the sample variance-covariance matrix in repeated measurement designs. (Submitted to Sankhyā).
[3] C. R. Rao: Linear Statistical Inference and Its Applications. J. Wiley, N. York 1965. MR 0221616 | Zbl 0137.36203
[4] C. R. Rao S. K. Mitra: Generalized Inverse of Matrices and Its Applications. J. Wiley, N. York 1971. MR 0338013
[5] R. Thrum J. Kleffe: Inequalities for moments of quadratic forms with applications to a.s. convergence. Math. Operationsforsch. Statistics Ser. Statistics (in print). MR 0704788
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