Summary: The research described in this paper began with the goal to develop critical tests of the two-high-threshold model of memory. Based on initial findings by D. Kellen and K. C. Klauer [“Discrete-state and continuous models of recognition memory: testing core properties under minimal assumptions”, J. Exp. Psychol., Learn. Mem. Cogn. 40, No. 6, 1795–1804 (2014; doi:10.1037/xlm0000016)], we focus on a ranking task in which one target and \(n - 1\) foils are ranked in decreasing order of perceived likelihood of being the target. The resulting distribution of interest is the probability that the target is ranked in position \(k\). We show that the two-high-threshold model predicts that the hazard function of this distribution increases monotonically for \(k > 1\). We generalize this result to the case in which each foil has a different probability of being detected under the model. These results are compared with two alternative models for recognition memory. The research exploring these issues has led us to an investigation of the Poisson-binomial distribution, showing that it too has a monotonically increasing hazard. In our opinion this finding is novel and significant because the Poisson binomial distribution can have many other applications outside of memory research and outside of psychology. In addition it is proved that any memory theory for the ranking task, which produces rank probabilities that form a strictly log-concave sequence, predicts a monotonically increasing hazard function. The hazard function for the ranking task for a single experimental condition thus enables a critical test for a potentially large class of memory theories other than just the two-high-threshold model.