A probability sampling design is a probability functionp(s) on subsetss of [1, ..., i, ..., N∼. Let π ij denote the joint inclusion probability fori andj. The problem is to determine conditions under which a fixed size (n) sampling designp exists so that π ij ∝(x i −x j )2 for a vector of real numbersx=(x 1, ...,x N ), or equivalently, so that for some order of the coefficients √π ik =√π ij +√π jk . Some necessary conditions for the proportionality to hold are obtained, and it is conjectured that it is satisfied only in special circumstances.
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Liu, T., Thompson, M.E. A problem arising in finite population sampling theory. Aeq. Math. 29, 307–312 (1985). https://doi.org/10.1007/BF02189834
- probability sampling
- joint inclusion probabilities
AMS (1980) subject classification
- Primary 26D99