On Karlin's Conjecture for Random Replacement Sampling Plans

Abstract

In 1974 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 $\mathscr{C}_K$ (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 $\mathscr{C}_K$ is not suitable for A being implied by B, i.e. one direction of Karlin's conjecture is disproved.