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 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>