Efficiency comparisons for partially rank-ordered set sampling
Choose a set size \(m\). Draw \(M\) independent samples (sets) of size \(m\) and rank the units in each set, where ranking is by judgment or by using a covariate. A sample for measurement then consists of the smallest ranked unit in the first set, the second smallest ranked in the second set, and so on\dots This will be a balanced ranked-set sample. We shall now describe a balanced partially rank-ordered set sample (PROSS). Consider a vector of subset sizes \((m_1,m_2,\dots,m_M)\) with a total set size \(m=\sum^M_{i=1}m_i\). Draw \(m\) independent sets of size \(m\) and divide the units in each set into \(M\) ordered subsets of sizes \((m_1,m_2,\dots,m_M)\) such that for \(1\leq i\leq j\leq M\), all units in the \(i\)th subset are ranked smaller than all the units in the \(j\)th subset. Next, a sample for measurement is obtained by selecting one unit drawn at random from the first subset in each of the first \(m_1\) sets, one unit drawn at random from the second subset in each of the \(m_2\) sets, and so on. This gives a sample of \(m\) independent values, with \(m_i\) values coming from the \(i\)th subset in their set of size \(m\). It may be noted that the PROSS scheme \((1,1,\dots 1)\) where there are \(m\) subsets each of size 1 corresponds to balanced RSS with set size \(m\). In this paper, the authors study the efficiency of PROSS sample mean under perfect rankings for various such PROSS schemes. They show that PROSS \((km_1,km_2,\dots,km_M)\) is always more efficient than the scheme \((m_1,m_2,\dots,m_M)\) for \(k>1\). Furthermore, they give a comparison of two PROSS schemes in the two subset case. Finally, Ozturk's conjecture that PROSS with \(m\) subsets of size \(k>1\) is always more efficient than RSS with set size \(m\) is proved as a corollary to a more general result which uses majorization.
scientific article; zbMATH DE number 6818220
scientific article