Godambe and Joshi (1965) have shown that for any design the Horvitz-Thompson estimator is admissible in the class of all unbiased estimators of a finite population total. In this note we give a simple alternative proof of this result using induction on the size of the population.
"An Alternative Proof of the Admissibility of the Horvitz-Thompson Estimator." Ann. Statist. 1 (3) 577 - 579, May, 1973. https://doi.org/10.1214/aos/1176342427