Semi-parametric empirical likelihood inference on quantile difference between two samples with length-biased and right-censored data

Exploring quantile differences between two populations at various probability levels offers valuable insights into their distinctions, which are essential for practical applications such as assessing treatment effects. However, estimating these differences can be challenging due to the complex data often encountered in clinical trials. This paper assumes that right-censored data and length-biased right-censored data originate from two populations of interest. We propose an adjusted smoothed empirical likelihood (EL) method for inferring quantile differences and establish the asymptotic properties of the proposed estimators. Under mild conditions, we demonstrate that the adjusted log-EL ratio statistics asymptotically follow the standard chi-squared distribution. We construct confidence intervals for the quantile differences using both normal and chi-squared approximations and develop a likelihood ratio test for these differences. The performance of our proposed methods is illustrated through simulation studies. Finally, we present a case study utilizing Oscar award nomination data to demonstrate the application of our method.