Testing the assumption of independence of truncation time and failure time

For data subject to random censorship, it is well known that the assumption of independence between censoring time and failure time cannot be tested nonparametrically. For data subject to random truncation, however, it is possible to test independence of truncation time and failure time. A statistic generalizing Kendall's tau is proposed for testing the independence assumption, and asymptotic properties under the null hypothesis are derived. A conditional permutation test is also derived for small data sets. Generalization to allow right-censoring and an application to truncated linear regression are also briefly discussed.