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The analysis of left truncated and right censored data is very common in survival and reliability analysis. In lifetime studies patients are often subject to left truncation in addition to right censoring. For example, in bone marrow transplant studies based on International Bone Marrow Transplant Registry (IBMTR), the patients who die while waiting for the transplants will not be reported to the IBMTR. In this paper, we develop novel U-statistics under left truncation and right censoring. We prove the-consistency of the proposed U-statistics. We derive the asymptotic distribution of the U-statistics using the counting process technique. As an application of the U-statistics, we develop a simple non-parametric test for testing the independence between time to failure and cause of failure in competing risks when the observations are subject to left truncation and right censoring. The finite sample performance of the proposed test is evaluated through a Monte Carlo simulation study. Finally, we illustrate our test procedure using the lifetime data of transformers.

U-Statistics for left truncated and right censored data