Linear sampling estimations of sums

Linear estimates of the form $\displaystyle\widehat Z=\sum_i\alpha_iX_i$ for the sums $\displaystyle Z =\sum_iX_i$ are considered, where $X_i$, $i=1\div n$ are unknown constants and $\alpha$ is a random vector in $R^n$. Various classes of estimates are distinguished by requirements on the distribution law of $\alpha$. The estimates subordinate to the sampling plan are introduced, the latter is defined by the vector $\varepsilon=(\varepsilon_1,\dots,\varepsilon_n)\ \varepsilon_i = 0$ or $1$.
For symmetric sampling plans among the corresponding subordinate estimates the optimal ones having the minimal variance are found. The estimates near to optimal are obtained for the independent trials scheme. Two estimates are also considered for independent trials in nonsymmetric case and their asymptotic normality is proved.