A problem arising in finite population sampling theory (Q1069577)

A probability sampling design is a probability function p(s) on subsets s of \(\{\) 1,...,i,...,N\(\}\). Let \(\pi_{ij}\) denote the joint inclusion probability for i and j. The problem is to determine conditions under which a fixed size (n) sampling design p exists so that \(\pi_{ij}\propto (x_ i-x_ j)^ 2\) for a vector of real numbers \(x=(x_ 1,...,x_ N)\), or equivalently, so that for some order of the coefficients \(\sqrt{\pi_{ik}}=\sqrt{\pi_{ij}}+\sqrt{\pi_{jk}}\). Some necessary conditions for the proportinality to hold are obtained, and it is conjectured that it is satisfied only in special circumstances.