U-Statistics for Left Truncated and Right Censored Data

TL;DR

This paper introduces novel U-statistics tailored for analyzing survival data with both left truncation and right censoring, providing theoretical properties, a non-parametric independence test, and practical applications.

Contribution

The paper develops new U-statistics for complex survival data, proves their asymptotic properties, and creates a non-parametric test for independence under truncation and censoring.

Findings

U-statistics are $ oot n$-consistent.

Derived asymptotic distribution using counting process techniques.

Validated the test with Monte Carlo simulations and transformer data.

Abstract

The analysis left truncated and right censored data is very common in survival and reliability analysis. In lifetime studies patients 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 $\sqrt{n}$-consistency of the proposed U-statistics. We derive the asymptotic distribution of the U-statistics using 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 Monte Carlo simulation study. Finally we illustrate our test procedure using lifetime data of transformers.