Two sample rank tests under a random truncation model

We consider rank tests for the situation where observations are truncated and the probabilities of truncation depends on the value of the observation. Provided the same, but unknown, truncation function is operating on both populations, we can obtain distribution free rank tests for equality versus a location or scale alternative in the underlying populations. The large sample distributions are derived for these conditional tests and asymptotic relative efficiencies obtained. We apply these methods to the analysis of liver tumor data.