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Summary:
Let $X_1,\ldots,X_N$ be a finite random sequence with the expectation $EX_t=\alpha\varphi_t(1\leq t\leq N)$ and with the regular covariance matrix $\bold G$. The matrix $\bold G$ and the values of $\varphi_t$ are supposed to be known; $\alpha$ is an unknown parameter. The least squares estimate $\hat{\alpha}$ and the best linear unbiased estimate (BLUE) $\tilde{\alpha}$ of the parameter $\alpha$ are mentioned. The efficiency $\ell_N=var\ \hat{\alpha}/var\ \tilde{\alpha}$ is derived. The exact value of $\ell_N$ is given for cases when $X_1,\ldots,X_N$ is a finite part of the autoregressive series of the first and of the second order and $\varphi_t\equiv 1$ and $\varphi_t =t\ (1 \leq t\leq N)$ and for the autoregressive series of the $n$-th order with $\varphi_t\equiv 1$. The efficiency and the asymptotic efficiency of the BLUE $\tilde{\alpha}$ in cases when $\bold G$ is not true covariance matrix is also considered.
References:
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