Empirical Likelihood Inference of the Buckley–James Estimator with Length-Biased Data

A problem that statisticians frequently face is the analysis of survival data obtained from a non-random sampling procedure. When each subject can be selected with a chance proportional to its measure, the bias imposed on the sample is called length bias. This paper uses empirical likelihood to construct confidence intervals for the regression coefficients using a Buckley–James type estimator when the underlying sample is length-biased. For this purpose, the empirical log-likelihood ratio is derived, and its asymptotic distribution is shown to be a standard chi-square. A simulation study is carried out to compare the confidence intervals based on the empirical likelihood and those based on the normal approximation. Following this, it is revealed that the empirical likelihood method improves the performance of the confidence intervals, specifically for small sample sizes. Finally, the methods are illustrated by modeling the regression parameter and estimating confidence intervals for a set of real data.