Minima, Zero-points, Saddle-points and Singularities of the Variance of the Horvitz-Thompson Estimator in Selection With Unequal Probabilities Without Replacement / Minima, Nullpunkte, Sattelpunkte und Singularitäten der Varianz des Horvitz-Thompson Schätzers bei Auswahl mit unterschiedlichen Wahrscheinlichkeiten ohne Zurücklegen

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 the sample. If y N /Y > 1/n, units must be taken apriori into the sample to attain V = 0. An objective rule for the number of these units is given.