Summary It is shown that a minimum of the variance V(Ŷ HT ) of the Horvitz-Thompson estimator with value zero can be found in the interior of the region of feasible probabilities of selection z i (0 < z i < 1; ∑ z i = 1), if y N /Y < 1/n, where the units i = 1,...,N of the population are ordered according to their values y i > 0; Y is the total. Other zero-points of the variance can be found by taking units apriori into...</z>

Summary With Efron’s law-school example the bootstrap method is compared with an alternative method, called doubling. It is shown, that the mean deviation of the estimator is always smaller for the doubling method.

Summary In this paper the variance of the Horvitz-Thompson estimator is minimized by an iterative method with the help of spline functions. As result an objective rule is given, how many of the units with greatest size must be taken apriori into the sample.