Let pii be the inclusion probability of unit i in a certain

Let (pi)i be the inclusion probability of unit i in a certain sample scheme (not always SRS) for drawing n units out of a population of size N. Prove that Sum[(pi)i]=n.

Solution

On a Scheme of Sampling of Two Units With Inclusion Probability Proportional to Size

Abstract: This paper introduces an unequal probability sampling without replacement scheme with inclusion probability proportional to size. This new scheme possesses some desirable properties with regard to ?i and ?ij , and provides a non-negative variance estimator of the Horvitz and Thompson estimator, when the values of the auxiliary variable fulfill some restrictions. On comparing the suggested scheme with some of the existing sampling schemes in respect of efficiency and stability of the variance estimator empirically, it has been observed that the performance of the scheme is satisfactory. Zusammenfassung: Wir stellen hier einen Stichprobenplan vor, der ohne Wiederholung und mit Aufnahmewahrscheinlichkeit proportional zum Umfang arbeitet. Dieser neue Plan weist bzgl. ?i und ?ij einige wunschenswerte