A New Approach to Sampling from Finite Populations. I Sufficiency and Linear Estimation

Summary Barnard (1963) introduced the concept of linear sufficiency in connection with the Gauss-Markov set-up of estimation. In the present article the concept of linear sufficiency is redefined to suit the problem of estimation in survey sampling where, as demonstrated by the present author (Godambe, 1955, 1965), the Gauss-Markov set-up is fundamentally inapplicable. According to this redefined concept, a certain estimator is shown to be uniquely (up to a constant multiplier), linearly sufficient for the population total in the entire class (defined by the author (1955)) of linear estimators. The significance of this result will be clear on the background of the author's previous result (Godambe, 1955) demonstrating the non-existence of a uniformly minimum variance estimator in the entire class of unbiased linear estimators for the population total. Practically, the results are summarized in Theorems 5.1 and 6.3.