Abstract
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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References
Godambe, V. P.,A unified theory of sampling from finite populations. J. Roy. Statist. Soc. Ser. B17 (1955), 269–278.
Godambe, V. P. andJoshi, V. M.,Admissibility and Bayes estimation in sampling finite populations I. Ann. Math. Statist.36 (1965), 1707–1722.
Hanif, M. andBrewer, K. R. W.,Sampling with unequal probabilities without replacement: A review. Inter. Stat. Rev.48 (1980), 317–335.
Hanurav, T. V.,On the Horvitz and Thompson estimator. Sankyā Ser. A24 (1962), 429–436.
Horvitz, D. G. andThompson, D. J. A generalization of sampling without replacement from a finite universe. J. Amer. Stat. Assoc.47 (1952), 663–685.
Murthy, M. N.,Generalized unbiased estimation in sampling from finite populations. Sankya Ser. B25 (1963), 245–262.
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Liu, TP., Thompson, M.E. A problem arising in finite population sampling theory. Aeq. Math. 29, 307–312 (1985). https://doi.org/10.1007/BF02189834
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DOI: https://doi.org/10.1007/BF02189834