On Karlin's conjecture for random replacement sampling plans
In ibid. 2, 1065-1094 (1974; Zbl 0306.62004) \textit{S. Karlin} introduced the concept of random replacement schemes and conjectured that the componentwise monotonicity of the replacement probabilities (condition A) is equivalent to a corresponding ordering of expectations of all functions \(\phi\) from a certain class \({\mathcal C}_ K\) (condition B). In this paper it is shown that A implies B for sample sizes \(n\leq 5\) and - provided the sample space is sufficiently large - also for \(n\geq 6\). By a counterexample it is shown that \({\mathcal C}_ K\) is not suitable for A being implied by B, i.e. one direction of Karlin's conjecture is disproved.
combinatorial inequalities, partial ordering
scientific article; zbMATH DE number 3965166